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+Improved Differential-neural Cryptanalysis for
+Round-reduced Simeck32/64 ∗
+Liu Zhang1,3[0000−0001−6106−3767], Jinyu Lu2(�)[0000−0002−7299−0934],
+Zilong Wang1,3[0000−0002−1525−3356], and Chao Li2,3[0000−0001−7467−7573]
+1 School of Cyber Engineering, Xidian University, Xi’an 710126, China
+{liuzhang@stu., zlwang@}xidian.edu.cn
+2 College of Sciences, National University of Defense Technology, Hunan, Changsha
+410073, China, [email protected], [email protected]
+3 State Key Laboratory of Cryptology, P.O.Box 5159, Beijing 100878, China
+Abstract. In CRYPTO 2019, Gohr presented differential-neural crypt-
+analysis by building the differential distinguisher with a neural network,
+achieving practical 11-, and 12-round key recovery attack for Speck32/64.
+Inspired by this framework, we develop the Inception neural network that
+is compatible with the round function of Simeck to improve the accuracy
+of the neural distinguishers, thus improving the accuracy of (9-12)-round
+neural distinguishers for Simeck32/64. To provide solid baselines for neu-
+ral distinguishers, we compute the full distribution of differences induced
+by one specific input difference up to 13-round Simeck32/64. Moreover,
+the performance of the DDT-based distinguishers in multiple ciphertext
+pairs is evaluated. Compared with the DDT-based distinguishers, the 9-,
+and 10-round neural distinguishers achieve better accuracy. Also, an in-
+depth analysis of the wrong key response profile revealed that the 12-th
+and 13-th bits of the subkey have little effect on the score of the neu-
+ral distinguisher, thereby accelerating key recovery attacks. Finally, an
+enhanced 15-round and the first practical 16-, and 17-round attacks are
+implemented for Simeck32/64, and the success rate of both the 15-, and
+16-round attacks is almost 100%.
+Keywords: Neural Distinguisher, Wrong Key Response Profile, Key
+Recovery Attack, Simeck32/64
+1
+Introduction
+Lightweight block ciphers present trade-offs between appropriate security and
+small resource-constrained devices, which is an essential foundation for data con-
+fidentiality in resource-constrained environments. Therefore, the design require-
+ments and security analysis of lightweight block ciphers are of great importance.
+Combining traditional analysis methods with “machine speed” to efficiently and
+∗ Supported by organization x.
+First Author and Second Author contribute equally to this work.
+arXiv:2301.11601v1  [cs.CR]  27 Jan 2023
+
+intelligently evaluate the security of cryptographic algorithm components, is one
+of the critical points and trends of current research. The development of Artificial
+Intelligence (AI) provides new opportunities for cryptanalysis.
+In CRYPTO 2019 [8], Gohr creatively combines deep learning with differ-
+ential cryptanalysis and applies it to the Speck32/64, gaining the neural dis-
+tinguisher (ND) can surpass the DDT-based distinguisher (DD). Then, a hy-
+brid distinguisher (HD) consisting of a ND and a classical differential (CD)
+with highly selective key search strategies result in forceful practical 11-, and
+12-round key recovery attacks. In EUROCRYPT 2021 [7], Benamira et al. pro-
+posed a thorough analysis of Gohr’s neural network. They discovered that these
+distinguishers are basing their decisions on the ciphertext pair difference and the
+internal state difference in penultimate and antepenultimate rounds.
+To attack more rounds, the component CD or ND must be extended. In
+ASIACRYPT 2022
+[4], Bao et al. devised the first practical 13-round and an
+improved 12-round ND-based key recovery attacks for Speck32/64 by enhanc-
+ing the CDs, which they deeply explored more generalized neutral bits of dif-
+ferentials, i.e., conditional (simultaneous) neutral bit/bit-sets. In addition, they
+obtained NDs up to 11-round Simon32/64 by using DenseNet and SENet, thus
+launching the practical 16-round key recovery attack. Zhang et al. [16] focused
+on improving the accuracy of ND and added the Inception composed of the
+multiple-parallel convolutional layers before the Residual network to capture
+information on multiple dimensions. Under the combined effect of multiple im-
+provements, they reduced the time complexity of key recovery attacks for 12-,
+and 13-round Speck32/64 and 16-round Simon32/64. They also devised the
+first practical 17-round key recovery for Simon32/64.
+The Simeck algorithm [15], which combines the good design components
+from both Simon and Speck [5] designed by National Security Agency (NSA),
+has received a lot of attention for its security. In 2022, Lyu et al. [13] improved
+Gohr’s framework and applied it to Simeck32/64. They obtained (8-10)-round
+NDs for Simeck32/64 and successfully accomplished attacks for (13-15)-round
+Simeck32/64 with low data complexity and time complexity. In the same year,
+Lu et al. [12] adopted the multiple ciphertext pairs (8 ciphertext pairs) to train
+the SE-ResNet neural network fed with a new data format for Simon and
+Simeck. Finally, they obtained (9-12)-round NDs for Simeck32/64. This raises
+the question of whether the key recovery attack for Simeck can be enhanced.
+Our Contribution. The contributions of this work are summarized as follows.
+• We improved the Inception neural network proposed by zhang et al. [16] ac-
+cording to the number of cyclic rotation in the round function of Simeck32/64.
+Meanwhile, to capture the connections between ciphertext pairs, we use mul-
+tiple ciphertext pairs forming a sample as the input of the neural network.
+Therefore, we improved the accuracy of (9-12)-round NDs using the ba-
+sic training method and staged training method. The result can be seen in
+Table 3.
+
+• To provide solid baselines for NDs, the full distribution of differences induced
+by the input difference (0x0000, 0x0040) is computed up to 13 rounds for
+Simeck32/64. Also, to make a fair comparison with NDs, the accuracy of
+the DDs with multiple ciphertext pairs under independent assumptions is
+investigated. The comparison shows that the 9-, and 10-round NDs achieve
+higher accuracy than the DDs, i.e., the ND contains more information than
+the DDs (see Table 3).
+• Based on the wrong key random hypothesis, we computed the score of the
+ND for ciphertexts decrypted with different wrong keys and derived the
+wrong key response profile (see Figure 3). Through a thorough study of the
+wrong key response profile, we found that the 12-th and 13-th bit subkeys
+have little effect on the score of the ND, but the ND is extremely sensitive
+to the 14-th, and 15-th bit subkeys. Thus optimizing the Bayesian key search
+algorithm (see Algorithm 3) and accelerating the key recovery attack.
+• We enhanced the 15-round and launched the first practical 16-, 17-round
+key recovery attacks for Simeck32/64 based on the ND. Table 1 provides a
+summary of these results.
+Table 1. Summary of key recovery attacks on Simeck32/64
+Attacks
+R
+Configure
+Data
+Time
+Success Rate
+Ref.
+ND
+13
+1+2+9+1
+216
+227.95+5⋆
+88%
+[13]
+14
+1+3+9+1
+223
+232.99+5⋆
+88%
+[13]
+15
+1+3+10+1
+224
+233.90+5⋆
+88%
+[13]
+1+3+10+1
+222
+235.309
+99.17%
+Sect. 5
+16
+1+3+11+1
+224
+238.189
+100%
+Sect. 5
+17
+1+3+12+1
+226
+245.037
+30%
+Sect. 5
+1.
+⋆: Time complexity is calculated in terms of the number of full rounds of
+Simeck32/64 encryption per second of 223.304 in [13]. For a fair comparison, we
+convert the time complexity to be calculated in terms of the number of 1-round
+decryption performed per second. These two benchmarks differ by about 25.
+2. Time complexity is calculated based on that one-second equals to 226.693 1-round
+decryption per second in this paper. Also, 221.762 full-rounds of Simeck32/64 en-
+cryption per second can be performed on our device.
+Organization. The rest of the paper is organized as follows. Section 2 introduces
+the design of Simeck and gives the preliminary on the ND model. Section 3
+gives the data format, network structure, training method, and result of NDs
+The
+experiment
+is
+conducted
+by
+Python
+3.7.15
+and
+Tensorflow
+2.5.0
+in
+Ubuntu
+20.04.
+The
+device
+information
+is
+Intel
+Xeon
+E5-2680V4*2
+with
+2.40GHz,
+256GB
+RAM,
+and
+NVIDIA
+RTX3080Ti
+12GB*6.
+The
+source
+code
+is
+available
+on
+GitHub
+https://github.com/CryptAnalystDesigner/
+Differential-Neural-Cryptanalysis-Simeck32.git.
+
+for Simeck32/64. Section 4 describes the neutral bits and wrong key response
+profiles used for key recovery attacks. Section 5 exhibits details of the (15-17)-
+round key recovery attacks. Section 6 concludes this paper.
+2
+Preliminary
+In this paper, we denote an n-bit binary vector by x = (xn−1, . . . , x0), where
+xi is the bit in position i with x0 the least significant one. ⊕ and ⊙ denote the
+eXclusive-OR operation and the bitwise AND operation, respectively. x ≪ γ
+or Sγ(x) represent circular left shift of x by γ bits. x ≫ γ or S−γ(x) represent
+circular right shift of x by γ bits. x ∥ y represents the concatenation of bit strings
+x and y.
+2.1
+A Brief Description of Simeck
+The Simeck family of lightweight block cipher was designed by Yang et al. in
+CHES 2015 [15]. To develop even more compact and efficient block ciphers, it
+incorporates good design components from both Simon and Speck designed by
+NSA. A standardized approach for lightweight cryptography was proposed by
+the National Institute of Standards and Technology (NIST) in 2019. Some ideas
+for this project use modified Simeck as a fundamental module, such as ACE [1],
+SPOC [2], and SPIX [3], which suggests that Simeck has more practical promise.
+Simeck adopt the feistel structure to perform encryptions or decryptions on
+2n-bit message blocks using a 4n-bit key, while n is the word size. The round
+function of Simeck is defined as f5,0,1(x) =
+�
+S5 (x) ⊙ x
+�
+⊕ S1(x). Designers
+reuse the round function in the key schedule to subkeys like Speck does. The
+encryption algorithm of Simeck32/64 is listed in Algorithms 1.
+Algorithm 1: Encryption of Simeck32/64.
+Input: P = (x0, y0): the paintext, (k0, k1, �� · · , k31): the round keys.
+Output: C = (x32, y32): the ciphertext.
+1 for r = 0 to 31 do
+2
+xr+1 ← (xr ≪ 5) & xr ⊕ (xr ≪ 1)
+3
+yr+1 ← xr
+4 end
+2.2
+Overview of Neural Distinguisher Model
+The ND is a supervised model which distinguishes whether ciphertexts are en-
+crypted by plaintexts that satisfies a specific input difference or by random
+numbers. Given m plaintext pairs {(Pi,0, Pi,1), i ∈ [0, m − 1]} and target cipher,
+the resulting ciphertext pairs {(Ci,0, Ci,1), i ∈ [0, m−1]} is regarded as a sample.
+Each sample will be attached with a label Y :
+Y =
+�1, if Pi,0 ⊕ Pi,1 = ∆, i ∈ [0, m − 1]
+0, if Pi,0 ⊕ Pi,1 ̸= ∆, i ∈ [0, m − 1]
+
+A large number of samples are fed into the neural network for training. Then,
+the ND model can be described as:
+Pr(Y = 1 | X0, . . . , Xm−1) = F (f(X0), · · · , f(Xm−1), ϕ(f(X0), · · · , f(Xm−1))) ,
+Xi = (Ci,0, Ci,1), i ∈ [0, m − 1],
+Pr(Y = 1 | X0, · · · , Xm−1) ∈ [0, 1],
+where f(Xi) represents the basic features of a ciphertext pair Xi, ϕ(·) is the
+derived features, and F(·) is the new posterior probability estimation function.
+3
+Neural Distinguisher for Simeck32/64
+It is crucial that a well-performing ND be obtained before a key recovery
+can be conducted. In this section, we provided the state-of-the-art NDs for
+Simeck32/64. More importantly, the DDs resulting from the input difference
+(0x0000, 0x0040) are computed up to 13 rounds for Simeck32/64. These DDs
+provide a solid baseline for NDs.
+3.1
+Construction of the Dataset
+Data quality is fundamentally the most important factor affecting the good-
+ness of a model. Constructing a good dataset for NDs requires answering the
+following questions:
+• How to select a good input difference?
+• What data format is used for a sample?
+• How many ciphertext pairs are contained in a sample?
+Input Difference. Numerous experiments have shown that the input difference
+has a significant impact on the accuracy of the NDs/DDs [4,6,7,8,9,10,13,14].
+Simultaneously, obtaining better results for the key recovery attack depends on
+whether the input difference of the NDs leads to better accuracy, while leading
+to the prepended CDs with high probability. Therefore, it is also necessary to
+consider the number of rounds and the neutral bits of the prepended CDs.
+The choice of input difference of NDs varies depending on the block cipher.
+For Simeck32/64, Lyu et al. [13] present two methods to select the input differ-
+ence of the NDs. In the first method, the input difference for the NDs is selected
+from the input difference of the classical differential trail of existing literature.
+As part of the second method, the MILP model was used to find input differ-
+ences for classical differential transitions that had high probabilities, then NDs
+based on these input differences were trained with short epochs, and then the
+NDs whose input differences had higher accuracy were selected for training long
+epochs. But they did not consider the effect of the Hamming weight of the input
+difference on the neural network. Lu et al. [12] studied the effect of the input
+difference of NDs of Hamming weight less than or equal to 3 on the performance
+of HDs, and their experiments showed that the input difference (0, ei) is a good
+
+choice to obtain a HD for Simon-like ciphers. Eventually, they built NDs for
+Simeck32/64 up to 12 rounds with input difference (0x0000, 0x0040).
+In this paper, we further explore the neutral bit of the input difference
+(0x0000, 0x0040) (see Sect. 4.1) and, in a comprehensive comparison, chose this
+input difference.
+Data Format. In the process of training a ND, the format of the sample needs
+to be specified in advance. This format is referred to as the ND’s data format
+for convenience. The most intuitive data format is the ciphertext pair (C, C′) =
+(xr, yr, x′
+r, y′
+r), which is used in Gohr’s network for Speck32/64 in [8,9]. As the
+research progressed, Benamira et al. [7] constructed a new data format (xr ⊕
+x′
+r, xr ⊕x′
+r ⊕yr ⊕y′
+r, xr ⊕yr, x′
+r ⊕y′
+r) through the output of the first convolution
+layer of Gohr’s neural network for Speck32/64, where xr ⊕ x′
+r represents the
+left branch difference of the ciphertext, xr ⊕ x′
+r ⊕ yr ⊕ y′
+r represents the right
+branch difference after decrypting one round of ciphertexts without knowing the
+(r − 1)-th subkey according to the round function of Speck, xr ⊕ yr/x′
+r ⊕ y′
+r
+represents the right branch ciphertext C/C′ of the penultimate round. It shows
+that the data format is closely related to the structure of the ciphers.
+Bao et al. [4] accepted data of the form (xr−1, x′
+r−1, yr−1 ⊕ y′
+r−1) for Si-
+mon32/64. Since when the output of the r-th round (C, C′) = (xr, yr, x′
+r, y′
+r)
+is known, one can directly compute (xr−1, x′
+r−1, yr−1 ⊕ y′
+r−1) without knowing
+the (r−1)-th subkey according to the round function of Simon-like ciphers. Lu et
+al. [12] further proposed a new data format (∆xr, ∆yr, xr, yr, x′
+r, y′
+r, ∆yr−1, p∆yr−2)
+and obtained better performance. The details are illustrated in Fig. 1, and this
+data format is used in this paper due to its superiority.
+Using Multiple Ciphertext Pairs. Gohr et al. [9] showed that for a single
+ciphertext pair, only their differences may provide information for Simon. One
+option to surpass DDs is to use multiple ciphertext pairs simultaneously, us-
+ing dependencies between the pairs, especially if the key is fixed. Therefore, in
+order to surpass DDs, we use multiple ciphertext pairs for training, and the re-
+sults (Section 3) confirm that multiple ciphertext pairs indeed help to surpass
+DDs, albeit only in some rounds. One current trend in deep learning-assisted
+cryptanalysis is the employment of multiple ciphertext pairs per sample, and
+our results offer solid evidence in favor of this trend.
+The three questions above have been addressed, and the dataset can be gen-
+erated. Specifically, training and test sets were generated by using the Linux
+random number generator to obtain uniformly distributed keys Ki and mul-
+tiple plaintext pairs {(Pi,j,0, Pi,j,1), j ∈ [0, m − 1]} with the input difference
+(0x0000, 0x0040) as well as a vector of binary-valued labels Yi. During the pro-
+duction of the training or test sets for r-round Simeck32/64, the multiple plain-
+text pairs were then encrypted for r rounds if Yi = 1, while otherwise, the second
+plaintext of the pairs were replaced with a freshly generated random plaintext
+and then encrypted for r rounds. Then use the r-round ciphertext pairs to gen-
+erate samples with data of form (∆xr, ∆yr, xr, yr, x′
+r, y′
+r, ∆yr−1, p∆yr−2).
+
+∆xr−1 = ∆yr
+∆yr−1 = yr−1 ⊕ y
+′
+r−1
+Sa
+Sb
+Sc
+kr−1
+∆xr = xr ⊕ x
+′
+r
+∆yr = yr ⊕ y
+′
+r
+∆xr−2 = ∆yr−1
+p∆yr−2
+Sa
+Sb
+Sc
+kr−2
+Fig. 1. Notation of the data format for Simon-like ciphers, where yr−1 = Sa(yr) ⊙
+Sb(yr)⊕Sc(yr)⊕xr ⊕kr−1 ≜ A⊕kr−1, y
+′
+r−1 = Sa(y
+′
+r)⊙Sb(y
+′
+r)⊕Sc(y
+′
+r)⊕x
+′
+r ⊕kr−1 ≜
+A
+′ ⊕ kr−1, and p∆yr−2 = Sa(A) ⊙ Sb(A) ⊕ Sc(A) ⊕ yr ⊕ Sa(A
+′) ⊙ Sb(A
+′) ⊕ Sc(A
+′) ⊕ y
+′
+r
+3.2
+Network Architecture
+In CRYPTO 2019, Gohr [8] used the Residual Network to capture the dif-
+ferential information between the ciphertext pairs, thus getting the ND for
+Speck32/64. To learn the XOR relation at the same position of the cipher-
+text, a one-dimensional convolution of kernel size 1 is used in Gohr’s network
+architecture. Since there may be some intrinsic connection between several adja-
+cent bits, Zhang et al. [16] added multiple one-dimensional convolutional layers
+with different kernel sizes in front of the residual block according to the circular
+shift operation in the round function of Speck32/64 and Simon32/64. In this
+paper, we improved Zhang et al.’s neural network to fit with the round function
+of Simeck to improve the accuracy of the NDs, the framework shown in Fig. 2.
+Initial Convolution (Module 1). The input layer is connected to the initial
+convolutional layer, which comprises two convolutional layers with Nf channels
+of kernel sizes 1 and 5. The two convolution layers are concatenated at the chan-
+nel dimension. Batch normalization is applied to the output of the concatenate
+layers. Finally, rectifier nonlinearity is applied to the output of batch normaliza-
+tion, and the resulting [m, ω, 2Nf] matrix is passed to the convolutional blocks
+layer where m = 8, ω = 16 and Nf = 32.
+Convolutional Blocks (Module 2). Each convolutional block consists of two
+layers of 2Nf filters. Each block applies first the convolution with kernel size
+
+Output
+Module 3
+Module 2
+Module 2
+Module 1
+Input
+F(·)
+f(·)
+Module 1
+Conv, 1, Nf
+Conv, 5, Nf
+Concatenate, 2Nf
+BN
+Relu
+Module 2
+ks = ks + 2
+Conv, ks, 2Nf
+BN
+Relu
+Conv, ks, 2Nf
+BN
+Relu
+⊕
+Module 3
+FC, d1
+BN
+Relu
+FC, d2
+BN
+Relu
+Output
+FC, 1
+Sigmod
+Fig. 2. The network architecture for Simeck32/64
+ks, then a batch normalization, and finally a rectifier layer. At the end of the
+convolutional block, a skip connection is added to the output of the final rec-
+tifier layer of the block to the input of the convolutional block. It transfers the
+result to the next block. After each convolutional block, the kernel size ks in-
+creases by 2 where ks = 3. The number of convolutional blocks is 5 in our model.
+Prediction Head (Module 3 and Output). The prediction head consists of
+two hidden layers and one output unit. The three fully connected layers comprise
+d1, d2 units, followed by the batch normalization and rectifier layers where d1 =
+512 and d2 = 64. The final layer consists of a single output unit using the
+Sigmoid activation function.
+3.3
+The Training method of Differential-Neural Distinguisher
+The accuracy is the most critical indicator reflecting the performance of the neu-
+ral distinguisher. The following training method was carried out to verify the
+performance of our NDs.
+Basic Training Scheme. We run the training for 20 epochs on the dataset for
+N = 2∗107 and M = 2∗106. We set the batch size to 30000 and used Mirrored-
+Strategy of TensorFlow to distribute it equally among the 6 GPUs. Optimization
+was performed against mean square error loss plus a small penalty based on L2
+weights regularization parameter c = 10−5 using the Adam algorithm [11]. A
+cyclic learning rate schedule was applied, setting the learning rate li for epoch i
+
+to li = α+ (n−i) mod (n+1)
+n
+·(β −α) with α = 10−4, β = 2×10−3 and n = 9. The
+networks obtained at the end of each epoch were stored, and the best network
+by validation loss was evaluated against a test set.
+Training using the Staged Train Method. We use several stages of pre-
+training to train an r-round ND for Simeck. First, we use our (r−1)-round dis-
+tinguisher to recognize (r − 3)-round Simeck with the input difference (0x0140,
+0x0080) (the most likely difference to appear three rounds after the input differ-
+ence (0x0000, 0x0040). The training was done on 2 ∗ 107 instances for 10 epochs
+with a cyclic learning rate schedule (2×10−3, 10−4). Then we trained the distin-
+guisher to recognize r-round Simeck with the input difference (0x0000, 0x0040)
+by processing 2 ∗ 107 freshly generated instances for 10 epochs with a cyclic
+learning rate schedule (10−4, 10−5). Finally, the learning rate was dropped to
+10−5 after processing another 2 ∗ 107 new instances for 10 epochs.
+3.4
+Compared Result
+We presented the state-of-the-art NDs for Simeck32/ 64. Meanwhile, we calcu-
+late the DDs for Simeck32/64 triggered by the input difference (0x0000, 0x0040)
+up to 13 rounds to give baselines for NDs (see Table 2). This is accomplished
+through the use of the frameworks of Gohr’s implementation for Speck32/64
+and Bao et al.’s implementation for Simon32/64. The calculation is feasible on
+Simeck32/64 but quite expensive. In fact, the calculation took about 939 core-
+days of computation time and yielded about 34 gigabytes of distribution data
+for each round, which was saved on disk for further studies.
+Table 2. Accuracy of the DDs for Simeck32/64 with input difference (0x0000, 0x0040).
+Combined means that the corresponding single pair distinguisher was used by combin-
+ing the scores under independence assumption. For this, 2×106 samples, each consisting
+of the given number of pairs m, were used to evaluating the accuracy.
+R
+m
+1
+2
+4
+8
+16
+32
+64
+128
+256
+7
+0.9040
+0.9765
+0.9936
+0.9996
+1.0
+1.0
+1.0
+1.0
+1.0
+8
+0.7105
+0.7921
+0.8786
+0.9518
+0.9907
+0.9995
+1.0
+1.0
+1.0
+9
+0.5738
+0.6097
+0.6590
+0.7221
+0.8011
+0.8848
+0.9554
+0.9919
+0.9998
+10
+0.5194
+0.5299
+0.5462
+0.5677
+0.5984
+0.6403
+0.6977
+0.7690
+0.8517
+11
+0.5044
+0.5068
+0.5109
+0.5176
+0.5247
+0.5364
+0.5530
+0.5761
+0.6085
+12
+0.5010
+0.5017
+0.5025
+0.5039
+0.5055
+0.5083
+0.5121
+0.5176
+0.5259
+13
+0.5002
+0.5001
+0.5007
+0.5009
+0.5012
+0.5016
+0.5032
+0.5039
+0.5086
+
+It is important to note that when multiple ciphertext pairs are used as a
+sample in the NDs, comparing the accuracy of the DDs computed with a single
+ciphertext pair as a sample is not fair. Actually, the accuracy of the DDs with
+multiple ciphertext pairs per sample can be calculated. This calculation is im-
+plicitly used by Gohr in [8], and later Gohr et al. [9] explicitly proposed rules for
+combining probabilities/distinguisher responses (see Corollary 2 in [9]). One can
+use this rule to explicitly convert a distinguisher for one ciphertext pair into one
+for an arbitrary number of ciphertext pairs. Algorithm 2 gives the pseudo-code
+for computing this distinguisher, and the results are shown in Table 2.
+Algorithm 2: Convert the DD for one ciphertext pair into one for an
+m number of ciphertext pairs.
+Input: DDT: the R round DDT table; N: the number of samples for single
+ciphertext pairs; m: the combined number of ciphertext pairs for one
+sample.
+Output: the combined Acc, TPR, TNR with m ciphertext pairs.
+1 Y ← {}
+2 for i = 1 to N do
+3
+Y[i ∗ m] ← random{0, 1}
+4
+for j = 1 to m − 1 do
+5
+Y[i ∗ m − j] ← Y[i ∗ m]
+6
+end
+7 end
+8 Randomly generate N ∗ m samples [x1, x2, · · · , xN∗m] according to Y
+9 Z ← {}
+10 for i = 1 to N ∗ m do
+11
+Z[i] ← DDT[xi]
+12 end
+13 Z ← Z / (Z+2−32)
+14 Z ← mean(Z.reshape(N,m), axis=1)
+15 predict_Y ← {}
+16 for i = 1 to N ∗ m do
+17
+if
+Z[i] > 0.5 then
+18
+predict_Y[i] ← 1
+19
+end
+20
+else
+21
+predict_Y[i] ← 0
+22
+end
+23 end
+24 calculate Acc, TPR, TNR based on (Y, predict_Y)
+25 return Acc, TPR, TNR
+/* In our experiments, N takes 220 when m no more than 210.
+*/
+In addition, r-round ND should be compared with (r − 1)-round DD. Since
+the data fed to r-round ND is the value of the ciphertext, one can directly com-
+
+pute the differences on (r − 1)-round outputs without knowing the subkey. The
+results are represented in Table 3, which shows that we improved the accuracy of
+the NDs for Simeck32/64. More importantly, it is able to surpass the accuracy
+of DDs for 9- and 10-round.
+Table 3. Comparison of NDs on Simeck32/64 with 8 ciphertext pairs as a sample.
+The input difference of ND/DD is (0x0000, 0x0040). *: The staged training method is
+used to train ND.
+R
+Attack
+Network
+Acc
+TPR
+TNR
+Ref.
+9
+DD
+DDT
+0.9518
+0.9604
+0.9433
+Sect. 3
+ND
+SE-ResNet
+0.9952
+0.9989
+0.9914
+[12]
+ND
+Inception
+0.9954
+0.9986
+0.9920
+Sect. 3
+10
+DD
+DDT
+0.7221
+0.7126
+0.7316
+Sect. 3
+ND
+SE-ResNet
+0.7354
+0.7207
+0.7501
+[12]
+ND
+Inception
+0.7371
+0.7165
+0.7525
+Sect. 3
+11
+DD
+DDT
+0.5677
+0.5416
+0.5940
+Sect. 3
+ND
+SE-ResNet
+0.5646
+0.5356
+0.5936
+[12]
+ND
+Inception
+0.5657
+0.5363
+0.5954
+Sect. 3
+ND
+Inception
+0.5666⋆
+0.5441
+0.5895
+Sect. 3
+12
+DD
+DDT
+0.5176
+0.4737
+0.5615
+Sect. 3
+ND
+SE-ResNet
+0.5146⋆
+0.4770
+0.5522
+[12]
+ND
+Inception
+0.5161⋆
+0.4807
+0.5504
+Sect. 3
+4
+Neutral bits and Wrong Key Response Profile
+In Sect. 3, we provided the state-of-the-art NDs for Simeck32/64, which use to
+perform better key recovery attacks in the following section. In [8], Gohr provides
+a framework of (1+s+r +1)-round key recovery attack (refer to Appendix A.1)
+consisting of three techniques to increase the success rate and speed up the at-
+tacks, where s is the length of the CD, and r is the length of the ND. Here is a
+description of these techniques.
+Neutral Bits. In the key recovery attack, multiple samples (formed into a ci-
+phertext structure) decrypted by the guessed subkey are predicted using the
+distinguisher. Then, the multiple scores are combined according to formula vk =
+�nb
+i=1 Zk
+i/1−Zk
+i as the final score of that guessed subkey to reduce the misjudgment
+rate of the ND. Since the CD suspended in front of the ND are probabilistic, re-
+sulting in sample entering the distinguisher not satisfying the same distribution.
+
+Multiple samples generated by neutral bits will have the same distribution. Also,
+the lower the accuracy of the distinguisher, the more neutral bits are needed.
+Priority of Ciphertext Structure. Spending the same amount of compu-
+tation on every ciphertext structure is inefficient. Gohr used a generic method
+(automatic exploitation versus exploration tradeoff based on Upper Confidence
+Bounds) to focus the key search on the most promising ciphertext structures.
+The priority score of each ciphertext structure is si = ωi
+max + √nc ·
+�
+log2(j)/ni
+where denote by ωi
+max the highest distinguisher score, ni the number of previous
+iterations in which the ith ciphertext structure, j the number of the current
+iteration and √nc the number of ciphertext structures available.
+Wrong Key Response Profile. The key search policy based on Bayesian Op-
+timization drastically reduces the number of trial decryptions. The basic idea
+of this policy is the wrong key randomization hypothesis. This hypothesis does
+not hold when only one round of trial decryption is performed, especially in a
+lightweight cipher. The expected response of the ND upon wrong-key decryp-
+tion will depend on the bitwise difference between the trial and real keys. This
+wrong-key response profile can be captured in a precomputation. Give some
+trial decryptions, the optimization step then trials to come up with a new set
+of candidate keys to try. These new candidate keys are chosen to maximize the
+probability of the observed distinguisher responses.
+4.1
+Exploring Neutral Bits
+To be able to attack more rounds with the ND, the CD is generally prepended
+in front of the ND. For the resulting HD used in the key recovery attack, it is
+not straightforward to aggregate enough samples of the same distribution fed to
+the ND due to the prepended CD. To overcome this problem, Gohr [8] used the
+neutral bits of the CD. The more neutral bits there are for the prepended CD, the
+more samples of the same distribution could be generated for the ND. However,
+generally, the longer the CD, the fewer the neutral bits. Finding enough neutral
+bits for prepending a long CD over a weak ND becomes a difficult problem for
+devising a key recovery to cover more rounds. To solve this problem, Bao et al.
+exploited various generalized NBs to make weak ND usable again. Particularly,
+they employed conditional simultaneous neutral bit-sets (CSNBS) and switching
+bits for adjoining differentials (SBfAD), which are essential for achieving efficient
+12-round and practical 13-round attacks for Speck32/64.
+Thus, the first part of the key recovery attack focuses on finding various
+types of neutral bits. Given a differential, in order to find the neutral bits, it is
+generally divided into two steps: firstly, collect enough conforming pairs (correct
+pairs); secondly, flip the target bits of the conforming pair, or flip all the bits
+contained in the target set of bits, and check the probability that the new plain-
+text pair is still the conforming pair.
+
+Finding SNBSs for 3-round Differential. For the prepended 3-round CD
+(0x0140, 0x0200) → (0x0000, 0x0040) on top of the NDs, one can experimen-
+tally obtain 14 deterministic NBs and 2 SNBSs (simultaneously complementing
+up to 4 bits) using an exhaustive search. Concretely, for the 3-round differential
+(0x0140, 0x0200) → (0x0000, 0x0040), (simultaneous-) neutral bits and bit-sets
+are [3], [4], [5], [7], [8], [9], [13], [14], [15], [18], [20], [22], [24], [30], [0, 31], [10, 25].
+Finding SNBSs for 4-round Differential. For the prepended 4-round CD
+(0x0300, 0x0440) → (0x0000, 0x0040) on top of the NDs, there are 7 com-
+plete NB/SNBS: [2], [4], [6], [8], [14], [9, 24], [9, 10, 25]. Still, the numbers of
+NBs/SNBSs are not enough for appending a weak neural network distinguisher.
+Thus, conditional ones were searched using Algorithm 3 in paper [4], and the
+obtained CSNBSs and their conditions are summarized together in Table 4.
+Table
+4.
+CSNBS
+for
+4-round
+Classical
+Differential
+(0x0300, 0x0440)
+→
+(0x0000, 0x0040) of Simeck32/64
+Bit-set
+C.
+Bit-set
+C.
+x[0, 10]
+x[2, 12]
+[21]
+00
+[23]
+00
+[21, 5]
+10
+[23, 12]
+10
+[21, 10]
+01
+[23, 7]
+01
+[21, 10, 5]
+11
+[23, 12, 7]
+11
+C.: Condition on x[i, j], e.g., x[i, j] = 10 means x[i] = 1 and x[j] = 0.
+4.2
+Wrong Key Response Profile
+To calculate the r-round wrong key response profile, we generated 3000 random
+keys and multiple input pairs {(Pi,0, Pi,1), i ∈ [0, m − 1]} for each difference
+δ ∈ (0, 216) and encrypted for r +1 rounds to obtain ciphertexts {(Ci,0, Ci,1), i ∈
+[0, m − 1]}, where Pi,0 ⊕ Pi,1 = ∆. Denoting the final real subkey of each
+encryption operation by k, we then performed single-round decryption to get
+E−1
+k⊕δ({Ci,0, i ∈ [0, m − 1]}), E−1
+k⊕δ({Ci,1, i ∈ [0, m − 1]}) and had the resulting
+partially decrypted ciphertext pair rated by an r-round ND. µδ and σδ were
+then calculated as empirical mean and standard deviation over these 3000 trials.
+We call the r-round wrong key response profile WKRPr. From the wrong key
+Response Profile, we can find some rules to speed up the key recovery attack.
+• Analysis of WKRP9. In Figure 3a, when the difference between guessed
+key and real key δ is greater than 16384, the score of the distinguisher is
+close to 0. This phenomenon indicates that the score of the distinguisher
+is very low when the 14-th and 15-th bit is guessed incorrectly. When δ ∈
+{2048, 4096, 8192, 10240, 12288, 14436}, the score of the distinguisher is greater
+
+than 0.6. This indicates that when the 11-th, 12-th, and 13-th bits are guessed
+incorrectly, it has little effect on the score of the distinguisher.
+• Analysis of WKRP10 and WKRP11. It is clear from Figure 3b that
+when the δ is greater than 32768, the score of the distinguisher is less than
+0.45, i.e., the 15-th bit has a greater impact on the distinguisher score. When
+δ ∈ {4096, 8192, 12288}, the score of the distinguisher is close to 0.55. This
+indicates that when the 12-th and 13-th bits are guessed incorrectly, it has
+little effect on the score of the distinguisher. It can also be observed from
+Figure 3c that the 12-th and 13-th bits have less influence on the score of
+the distinguisher, and the 14-th and 15-th bits have more influence on the
+score of the distinguisher.
+• Analysis of WKRP12. Despite the small difference in scores in Figure 3d,
+it was found that when only the 12-th and 13-th bits are wrongly guessed,
+the score of the distinguisher is still higher than the other positions.
+(a) WKRP9
+(b) WKRP10
+(c) WKRP11
+(d) WKRP12
+Fig. 3. Wrong Key Response Profile for Simeck32/64.
+
+1.0
+0.50
+0.8 
+Meanresponse
+0.6
+0.4 
+0.2 
+0.0
+0
+4096
+Differencetorealkey0.50
+0.65
+0.60
+0.55
+response
+0.50
+Mean
+0.45
+0.40 
+0.35
+0
+4096
+81921228816384204802457628672327683686440960450564915253248573446144065536
+Differenceto realkey0.520
+0.50
+0.515
+0.510
+0.505
+0.500
+Mean
+0.495
+0.490
+0.485
+0.480
+0
+Differenceto realkey0.50
+0.5015
+0.5010
+0.5005
+response
+0.5000
+Mean
+0.4995
+0.4990
+0.4985
+0.4980
+0.4975
+0
+4096
+81921228816384204802457628672327683686440960450564915253248573446144065536
+Differencetoreal keyFrom the four wrong key response profiles, we can conclude that when the
+14-th and 15-th bit subkeys are guessed incorrectly, it has a greater impact on
+the score of the distinguisher; when the 12-th and 13-th bit subkeys are guessed
+incorrectly, it has a smaller impact on the score of the distinguisher. According
+to these phenomena, we can speed up the key recovery attack.
+• Guess the 14-th and 15-th bit subkeys. Since the difference between
+the score of the distinguisher of bits 14 and 15 in the case of correct and
+incorrect guesses is relatively large, we can first determine the values of these
+two bits. Before performing a Bayesian key search, a random set of subkeys
+is guessed, then the 14-th and 15-th bits of the subkeys are traversed, and
+the ciphertext is decrypted using the subkeys. Thus, the values of the 14-th
+and 15-th bits can be determined based on the score of the distinguisher.
+The Bayesian key search algorithm can easily recover these two bits even if
+the values of these two bits are not determined in advance.
+• Ignore the 12-th and 13-th bit subkeys. Since the 12-th and 13-th bit
+subkeys have less influence on the score of the distinguisher, we first set
+these two bits to 0 when generating the first batch of candidate subkeys and
+then randomize the values of the two bits after completing the Bayesian key
+sorting and recommending the new candidate subkeys. Previous researchers
+have also exploited this feature to accelerate key recovery attacks, and the
+14-th and 15-th bit subkeys have little impact on the score of the distin-
+guisher when guessed incorrectly for Speck32/64 and Simon32/64[4,8,16].
+The Bayesian key search algorithm considering insensitive key bits is shown
+in Algorithm 3.
+5
+Practical Key Recovery Attack
+When a fast graphics card is used, the performance of the implementation is not
+limited by the speed of neural network evaluation but by the total number of
+iterations on the ciphertext structures. We count a key guess as successful if the
+sum of the Hamming weights of the differences between the returned last two
+subkeys and the real two subkeys are at most two. The experimental parameters
+for key recovery attacks are denoted as follows.
+1. ncts: the number of ciphertext structure.
+2. nb: the number of ciphertext pairs in each ciphertext structures.
+3. nit: the total number of iterations on the ciphertext structures.
+4. c1 and c2: the cutoffs with respect to the scores of the recommended last
+subkey and second to last subkey, respectively.
+5. nbyit1, ncand1 and nbyit2, ncand2: the number of iterations and number of key
+candidates within each iteration in the BayesianKeySearch Algorithm for
+guessing each of the last and the second to last subkeys, respectively.
+
+Algorithm 3: BayesianKeySearch Algorithm For Simeck32/64.
+Input: Ciphertext structure C := {C0, · · · , Cnb−1}, a neural distinguisher
+ND, and its wrong key response profile µ and σ, the number of
+candidates to be generated within each iteration ncand, the number of
+iterations nbyit
+Output: The list L of tuples of recommended keys and their scores
+1 S := {k0, k1, · · · , kncand−1} ← choose ncand values at random without
+replacement from the set of all subkey candidates
+2 S = S & 0xCFFF
+3 L ← {}
+4 for t = 1 to nbyit do
+5
+for ∀ki ∈ S do
+6
+for j = 0 to nb − 1 do
+7
+C
+′
+j,ki = F −1
+ki (Cj)
+8
+vj,ki = ND(C
+′
+j,ki)
+9
+sj,ki = log2(vj,ki/(1 − vj,ki))
+10
+end
+11
+ski = �nb−1
+j=0 sj,ki; /* the combined score of ki using neutral
+bits.
+*/
+12
+L ← L∥(ki, ski);
+13
+mki = �nb−1
+j=0 vj,ki/nb
+14
+end
+15
+for k ∈ {0, 1, · · · , 216 − 1} & 0xCFFF do
+16
+λk = �ncand−1
+i=0
+(mki − µki⊕k)2/σ2
+ki⊕k; /* using wrong key response
+profile.
+*/
+17
+end
+18
+S ← argsortk(λ)[0 : ncand − 1];
+19
+r := {r0, r1, · · · , rncand−1} ← choose ncand values at (0, 4) at random
+20
+r = r << 12; /* Randomize the 12-th and 13-th bit subkeys.
+*/
+21
+S = S ⊕ r
+22 end
+23 return L
+
+5.1
+Complexity Calculation
+Theoretical Data Complexity. The theoretical data complexity of the exper-
+iment is calculated by the formula nb × nct × m × 2. In the actual experiment,
+when the accuracy of the ND is high, the key can be recovered quickly and suc-
+cessfully. Not all the ciphertext structure is used, so the actual data complexity
+is lower than the theoretical.
+Experimental Time Complexity. The time complexity calculation formula
+in our experiments is 226.693 × rt × log1−sr 0.01, which is borrowed from [16].
+Our device can perform 226.693 1-round decryption per second. rt is the average
+running time of multiple experiments. The success rate sr is the number of suc-
+cessfully recovered subkeys divided by the number of experiments. We calculate
+how many experiments need to be performed to ensure at least one successful
+experiment. When the overall success rate is 99%, we consider the experiment
+to be successful, and the number of experiments ne is: 1−(1−sr)ne = 0.99, i.e.,
+log1−sr 0.01.
+5.2
+Key Recovery Attack on 15-round Simeck32/64
+Experiment 1: The components of key recovery attack ASimeck15R of 15-round
+Simeck32/64 are as follows.
+1. 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040).
+2. neutral bits of generating multiple ciphertext pairs: [3], [4], [5].
+3. neutral bits of combined response of neural distinguisher: [7], [8], [9], [13], [14],
+[15], [18], [20].
+4. 10-round neural distinguisher NDSimeck10R and wrong key response profiles
+NDSimon10R · µ and NDSimeck10R · δ.
+5. 9-round distinguisher NDSimeck9R and wrong key response profiles NDSimon9R·
+µ and NDSimeck9R · δ.
+Concrete parameters used in our 15-round key recovery attack ASimeck15R are
+listed as follows.
+m = 8
+nb = 28
+ncts = 210
+nit = 211
+c1 = 10
+c2 = 10
+nbyit1 = nbyit2 = 5
+ncand1 = ncand2 = 32
+The theoretical data complexity is m×nb ×ncts ×2 = 222 plaintexts. The ac-
+tual data complexity is 219.621. In total, 120 trials are running and 119 successful
+trials. Thus, the success rate sr is 99.17%. The average running time of the exper-
+iment rt is 407.901s. The time complexity is 226.693 × rt × log1−sr 0.01 = 235.309.
+5.3
+Key Recovery Attack on 16-round Simeck32/64
+Experiment 2: The components of key recovery attack ASimeck16R of 16-round
+Simeck32/64 are shown as follows.
+
+1. 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040).
+2. neutral bits of generating multiple ciphertext pairs: [3], [4], [5].
+3. neutral bits of combined response of neural distinguisher: [7], [8], [9], [13], [14],
+[15], [18], [20], [22], [24].
+4. 11-round neural distinguisher NDSimeck11R and wrong key response profiles
+NDSimeck11R · µ and NDSimeck11R · δ.
+5. 10-round neural distinguisher NDSimeck10R and wrong key response profiles
+NDSimeck10R · µ and NDSimeck10R · δ.
+Concrete parameters used in our 16-round key recovery attack ASimeck16R are
+listed as follows.
+m = 8
+nb = 210
+ncts = 210
+nit = 211
+c1 = 10
+c2 = 10
+nbyit1 = nbyit2 = 5
+ncand1 = ncand2 = 32
+The theoretical data complexity is m × nb × ncts × 2 = 224 plaintexts. The
+actual data complexity is 222.788. We use 6 processes, each running 20 experi-
+ments. Since the memory limit was exceeded during the experiment, one process
+was killed, leaving 100 experiments, 100 of which successfully recovered the key.
+Thus, the success rate sr is 100%. The average running time of the experiment
+rt is 2889.648s. The time complexity is 226.693 × rt = 238.189.
+5.4
+Key Recovery Attack on 17-round Simeck32/64
+Experiment 3: The components of key recovery attack ASimeck17R of 17-round
+Simeck32/64 are shown as follows.
+1. 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040).
+2. neutral bits of generating multiple ciphertext pairs: [3], [4], [5].
+3. neutral bits of combined response of neural distinguisher: [7], [8], [9], [13], [14],
+[15], [18], [20], [22], [24], [30], [0, 31].
+4. 12-round neural distinguisher NDSimeck12R and wrong key response profiles
+NDSimeck12R · µ and NDSimeck12R · δ.
+5. 11-round neural distinguisher NDSimeck11R and wrong key response profiles
+NDSimeck11R · µ and NDSimeck11R · δ.
+Concrete parameters used in our 17-round key recovery attack ASimeck17R are
+listed as follows.
+m = 8
+nb = 212
+ncts = 210
+nit = 211
+c1 = 20
+c2 = −120
+nbyit1 = nbyit2 = 5
+ncand1 = ncand2 = 32
+The theoretical data complexity is m × nb × ncts × 2 = 226 plaintexts. The
+actual data complexity is 225.935. In total, trials are 50 running, and there are
+15 successful trials. Thus, the success rate sr is 30%. The average running
+time of the experiment rt is 25774.822s. The time complexity is 226.693 × rt ×
+log1−sr 0.01 = 245.037.
+
+Remark 1. There are two reasons why we do not launch a 17-round key recovery
+attack using a 4-round CD and an 11-round ND. One is that the probability
+of the 4-round CD (0x0300, 0x0440) → (0x0000, 0x0040) is about 212 (the prob-
+ability of the 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040) is about 2−8),
+resulting in too much data required, and the second is that there are not enough
+neutral bits in the 4-round CD.
+6
+Conclusion
+In this paper, we show practical key recovery attacks up to 17 rounds of Simeck
+32/64, raising the technical level of practical attacks by two rounds. We design
+neural network that fits with the round function of Simeck to improve the ac-
+curacy of the neural distinguishers, and is able to outperform the DDT-based
+distinguisher in some rounds. To launch more rounds of the key recovery attack,
+we make a concerted effort on the classical differential and the neural distin-
+guisher to make both modules good. In addition, we optimize the key recovery
+attack process by deeply analyzing the wrong key response profile, thus reducing
+the complexity of the key recovery attack.
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+(2021)
+15. Yang, G., Zhu, B., Suder, V., Aagaard, M.D., Gong, G.: The simeck family of
+lightweight block ciphers. In: International Workshop on Cryptographic Hardware
+and Embedded Systems. pp. 307–329. Springer (2015)
+16. Zhang, L., Wang, Z., Wang, B.: Improving differential-neural cryptanalysis with
+inception blocks. Cryptology ePrint Archive (2022)
+
+A
+Appendix
+A.1
+Procedure of (1 + s + r + 1)-round key recovery attack
+The attack procedure is as follows.
+1. Initialize variables Gbestkey ← (None, None), Gbestscore ← −∞.
+2. Generate ncts random plaintext pairs with difference ∆P.
+3. Using ncts plaintext pairs and log2 m neutral bit with probability one to
+generate ncts multiple plaintext pairs. Every multiple plaintext pairs have
+m plaintext pairs.
+4. From the ncts multiple plaintext pairs, generate ncts plaintext structures
+using nb generalized neutral bit.
+5. Decrypt one round using zero as the subkey for all multiple plaintext pairs
+in the structures and obtain ncts plaintext structure.
+6. Query for the ciphertexts under (1 + s + r + 1)-round Simeck32/64 of the
+ncts × nb × 2 plaintext structures, thus obtain ncts ciphertext structures,
+denoted by {C1, . . . , Cncts}.
+7. Initialize an array ωmax and an array nvisit to record the highest distinguisher
+score obtained so far and the number of visits have received in the last subkey
+search for the ciphertext structures.
+8. Initialize variables bestscore ← −∞, bestkey ← (None, None), bestpos ←
+None to record the best score, the corresponding best recommended values
+for the two subkeys obtained among all ciphertext structures and the index
+of this ciphertext structures.
+9. For j from 1 to nit:
+(a) Compute the priority of each of the ciphertext structures as follows:
+si = ωmaxi + α ·
+�
+log2 j/nvisiti, for i ∈ {1, . . . , ncts}, and α = √ncts;
+The formula of priority is designed according to a general method in
+reinforcement learning for achieving automatic exploitation versus ex-
+ploration trade-off based on Upper Confidence Bounds. It is motivated
+to focus the key search on the most promising ciphertext structures [8].
+(b) Pick the ciphertext structure with the highest priority score for further
+processing in this j-th iteration, denote it by C, and its index by idx,
+nvisitidx ← nvisitidx + 1.
+(c) Run BayesianKeySearch Algorithm [8] with C, the r-round neural
+distinguisher NDr and its wrong key response profile NDr ·µ and NDr ·
+σ, ncand1, and nbyit1 as input parameters; obtain the output, that is a
+list L1 of nbyit1 × ncand1 candidate values for the last subkey and their
+scores, i.e., L1 = {(g1i, v1i) : i ∈ {1, . . . , nbyit1 × ncand1}}.
+(d) Find the maximum v1max among v1i in L1, if v1max > ωmaxidx, ωmaxidx ←
+v1max.
+(e) For each of recommended last subkey g1i ∈ L1, if the score v1i > c1,
+i. Decrypt the ciphertext in C using the g1i by one round and obtain
+the ciphertext structures C′ of (1 + s + r)-round Simeck32/64.
+
+ii. Run BayesianKeySearch Algorithm [8] with C′ , the neural dis-
+tinguisher NDr−1 and its wrong key response profile NDr−1 · µ and
+NDr−1·σ, ncand2, and nbyit2 as input parameters; obtain the output,
+that is a list L2 of nbyit2×ncand2 candidate values for the last subkey
+and their scores, i.e., L2 = {(g2i, v2i) : i ∈ {1, . . . , nbyit2 × ncand2}}.
+iii. Find the maximum v2i and the corresponding g2i in L2, and denote
+them by v2max and g2max.
+iv. If v2max > bestscore, update bestscore ← v2max, bestkey ← (g1i,
+g2max), bestpos ← idx.
+(f) If bestscore > c2, go to Step 10.
+10. Make a final improvement using VerifierSearch [8] on the value of bestkey
+by examining whether the scores of a set of keys obtained by changing at
+most 2 bits on top of the incrementally updated bestkey could be improved
+recursively until no improvement obtained, update bestscore to the best score
+in the final improvement; If bestscore > Gbestscore, update Gbestscore ←
+bestscore, Gbestkey ← bestkey.
+11. Return Gbestkey, Gbestscore.
+

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19FRT4oBgHgl3EQfmjee/content/tmp_files/2301.13602v1.pdf.txt

@@ -0,0 +1,1563 @@
+arXiv:2301.13602v1  [physics.flu-dyn]  31 Jan 2023
+Impact of an Arc-shaped Control Plate on Flow and
+Heat Transfer around a Isothermally Heated Rotating
+Circular Cylinder
+Amarjit Hatya, Rajendra K. Ray*a
+aSchool of Mathematical and Statistical Sciences, Indian Institute of Technology
+Mandi, Mandi, 175005, Himachal Pradesh, India
+Abstract
+The main objective of this paper is to study the flow characteristics of a rotat-
+ing, isothermally heated circular cylinder with a vertical arc-shaped control plate
+placed downstream. Stream function-Vorticity (ψ − ω) formulation of two di-
+mensional (2-D) Navier-Stokes (N-S) equations is considered as the governing
+equation and the simulations are performed for different distances of the control
+plate (0.5, 1, 2, 3), rotational rates (0.5, 1, 2.07, 3.25) at Prandtl number 0.7 and
+Reynolds number 150. The governing equations are discretized using the Higher
+Order Compact (HOC) scheme and the system of algebraic equations, arising
+from HOC discretization, is solved using the Bi-Conjugate Gradient Stabilized
+approach. Present computed results show that the vortex shedding plane is shifted
+upward from the centerline of the flow domain by the cylinder’s rotational mo-
+tion. The structure of the wake varies based on the plate’s position. The size of
+vortices is greatly reduced when the control plate is set at d/R0 = 3 and the rota-
+tional rate is very high. At greater rotational rates, the impact of varied positions
+of the arc-shaped control plate is very significant. The rotation of the cylinder and
+the location of the plate can be used to lower or enhance the values of drag and
+lift coefficients as well as the heat transfer from the surface of the cylinder. The
+maximum value of the drag coefficient, which is about 3, is achieved for d/R0 = 2
+and α = 3.25.
+Keywords: Navier-Stokes equations, Circular cylinder, Arc-shaped control plate,
+Heat transfer, HOC
+*Corresponding author: [email protected]
+
+1. Introduction
+Active control of flow past a rotating circular cylinder has always been an in-
+teresting topic in fluid dynamics. The wake behaviour for flow past a rotating
+cylinder is more complicated than for flow past a stationary cylinder because the
+rotation of the cylinder separates the shear layer and modifies the boundary layer.
+In 1928, Bickley [1] was among the first to attempt analytical study of the viscous
+flow over a rotating cylinder. He considered the potential flow created by a vortex
+in the vicinity of a cylinder. The wake structure in flow past a cylinder is compli-
+cated due to interactions between a boundary layer, a separating free shear layer,
+and a wake. It has huge significance in engineering as the alternating shedding
+pattern of the vortices in the wake causes considerable fluctuating pressure forces
+in a direction transverse to the fluid flow, which can produce structural vibrations,
+acoustic noise, or resonance, and in certain situations, structural collapse. In 1966,
+Gerrard [2] experimentally studied the flow past bluff bodies along with flow past
+circular cylinder with splitter plates for high Reynolds numbers. He found that
+the shear layer was drawn by the vortex formation from the opposite side of the
+wake across the center line of the wake, cutting off the vorticity supply to the ex-
+panding vortex. He found that the width of the gap between the cylinder and a
+splitter plate parallel to the flow, is the only relevant parameter than the position
+of the trailing edge of the plate. He studied the effect of a plate normal to the
+flow and found that the length of the effective vortex formation area equalled the
+distance of the plate from the domain boundary. He observed a substantial cross-
+flow velocity created near the plate when a vortex grew close behind it, facilitating
+the shedding process and increasing the frequency. Pralits et al. [3] numerically
+studied the flow past rotary cylinder and found that the increased rotational speed
+caused two distinct instability in the flow. Kang et al. [4] found that the vortex
+shedding was stopped completely when the cylinder rotation rate was set at twice
+the velocity of free stream fluid. Diaz et al. [5] have experimentally studied the
+flow past rotary cylinder for Reynolds number 9000. They saw a decrease in pe-
+riodic vortex activity and a rise in random modulation of the shedding process,
+which he attributed to the relocation of the stagnation point and the thickening of
+the spinning fluid layer near the cylinder surface. They discovered that when the
+rotating speed equals the free-stream speed, a regular periodic vortex shedding
+occurs, and that the periodic vortex shedding is suppressed at large velocity ra-
+tios. For velocity ratios equal to or greater than 1.5, they concluded that rotation
+considerably alters the traditional Karman vortex shedding. Similar findings were
+produced by Massons et al. [6] for flow past rotating cylinder. Stojkovic et al. [7]
+2
+
+studied the flow at greater rotation rates and discovered a second shedding mode
+in a limited interval [4.85, 5.15] of rotation rate where the shedding frequency was
+substantially lower than that of the traditional Von-Karman vortex shedding. At a
+high Reynolds number (Re = 105), Roshko [8] investigated the impact of a splitter
+plate positioned downstream of a bluff body and parallel to the free stream. By
+bringing the plate closer to the cylinder, he observed that the shedding frequency
+and base suction were reduced. Bearman [9] found that the separating shear flow
+on the top of the surface is pushed to rejoin if the circular cylinder with an end
+plate downstream is spun at a constant pace. As a result, the effects and vibrations
+caused by boundary-layer development are diminished, and the vortex formation
+is suppressed. Apelt et al. [10] used a horizontal splitter plate with varied lengths
+to diameter ratios less than 2 to investigate the flow past a circular cylinder for
+104 < Re < 5 × 104. The splitter plate considerably reduces drag by stabilising
+separation points, lowers the Strouhal number, and increases base pressure by
+roughly 50%, according to their research. They also discovered that when using
+a splitter plate instead of a cylinder without one, the wake pattern narrows. Kwon
+and Choi [11] indicated that there is a critical length of splitter plate that causes
+vortex shedding to totally disappear, and that this critical length is proportional
+to the Reynolds number. They also discovered that the Strouhal number rises as
+the plate’s length increases until it equals the cylinder’s diameter. Bao and Tao
+[12] analyzed the flow past a circular cylinder with twin parallel plates attached
+and discovered that optimal positioning can outperform the standard splitter plate.
+More studies with control plate can be found in [13–16].
+Along with studying the wake structure and pressure forces, force convective
+heat transfer from rotating cylinders has been widely investigated by many re-
+searchers for its many real-life applications and scientific interests. Drying cylin-
+drical items [17]; cylindrical cooling devices in the plastics and glass industries;
+drying and coating of papers using a hot spinning cylinder; chemical and food pro-
+cessing industries; textile and paper manufacturing, and so on are some examples
+of real-world uses. In an experiment, Anderson and Saunders [18] explored heat
+convection in a confined room filled with air using an isothermally heated rotating
+circular cylinder. Temperatures were elevated to 140 degrees Fahrenheit above the
+ambient temperature while air pressure was maintained at 4 atm. The experiment
+used three distinct cylinders, each with varying diameters (1, 1.8, and 3.9 inches)
+but the same length (2 feet). They determined that heat exchange is nearly steady
+when rotational speed is between 0 to a crucial value of 0.9, and past that point,
+heat exchange increases in proportion to the rotational speed’s 2/3 power. Badr
+3
+
+and Dennis [19] conducted a numerical research on force convective heat transfer
+from an unconfined rotary cylinder, concluding that increasing rotational speed re-
+duces overall rate of heat transfer because the cylinder is isolated from the stream
+by the spinning fluid layer. Mohanty et al. [20] performed experimental study on
+heat transfer from rotating cylinder for high Reynolds numbers. They discovered
+that rotational motion increases average heat transmission by roughly 30% when
+compared to a fixed cylinder with a fixed Reynolds number. They also discovered
+that as compared to stationary cylinders, rotational motion caused a lower heat
+transfer rate at the front stagnation point. An analytical study was attempted by
+Kendoush [21] and a formula, Nu = 0.6366(RePr)1/2, was proposed to compute
+the local Nusselt number (Nu) for low Prandtl numbers (Pr), where Re denotes
+the Reynolds number. With the help of the finite volume technique, Paramane and
+Sharma [22] studied the heat transfer and fluid flow across a rotating cylinder for
+Prandtl number of 0.7, low Reynolds numbers ranging from 20 to 160, and rotary
+speeds of 0 ≤ α ≤ 6. They discovered that when rotary speeds rise, the average
+Nusselt number falls while the Reynolds number rises. It was concluded that the
+rotation could be employed to reduce drag and suppress heat transmission from
+the cylinder. Sufyan et al. [23] discovered that low and medium rotary speeds
+immediately reduce heat transmission, but that at higher rotational rates, the in-
+creased size of the enclosing vortex causes even more heat transfer reduction. A
+few more studies on this subject can be found on [24–27].
+After an extensive literature survey, it is found that many researchers worked
+on heat transfer and flow across a rotating circular cylinder. There are numer-
+ous works on the flow across a circular cylinder with splitter plates and attached
+fins. Effect of curved fins and plates are studied by few researchers for missiles
+[28] and formula−1 cars [29] and these are being used in real life. There are
+some researchers who tried to study the wake structure and base pressure after
+applying the rotation to the cylinder with attached splitter plates or fins, but the
+effect of both rotation and the presence of control plates on the process of heat
+transfer is not tested. Considering the importance, the current investigation is cen-
+tred on the impact of a control plate on forced convective heat transfer and flow
+across a rotating circular cylinder. It can be useful in electronic equipment cool-
+ing and processing industries. We have taken into account an arc-shaped plate
+with a vertical orientation since we are considering a polar coordinate system
+with non-uniform grids. For this investigation, the Reynolds number is fixed at
+150 and the Prandtl number is fixed at 0.7. The plate distance to cylinder radius
+ratio varies between 0.5 and 3, while the rotational rates range from 0.5 to 3.25.
+4
+
+The two-dimensional unsteady Navier-Stokes equations and energy equation are
+first non-dimensionalized and then discretized by using a Higher Order Compact
+(HOC) scheme [30, 31] based on non-uniform polar grids. Temporal accuracy of
+2nd order and spatial accuracy of atleast 3rd order are obtained through the ap-
+plication of the finite difference scheme. To obtain a solution from a discretized
+system, the Bi-conjugate Gradient Stabilized method approach is employed.
+The paper is arranged as follows: in Section 2, we discuss the governing equa-
+tions and initial and boundary conditions related to the current problem; in Section
+3, the numerical scheme is described as well as the independence tests and valid-
+ity of the numerical scheme are produced; results are discussed in Section 4; and
+finally, we conclude our remarks in Section 5.
+2. The governing equations and the problem
+The considered system is represented in Fig. 1 as a two-dimensional unsteady,
+incompressible, laminar, and viscous flow of a Newtonian fluid over an isother-
+mally heated circular cylinder of radius R0. At ˆt = 0, the cylinder acquires the
+surface temperature Ts impulsively. The following formulas are used to transform
+dimensional parameters to dimensionless form: t = ˆtU∞
+R0 , r =
+ˆr
+R0, u =
+ˆu
+U∞, v =
+ˆv
+U∞,
+ψ = ˆψU∞
+R0 , ω = ˆωR0
+U∞ , φ = (T−T∞)
+(Ts−T∞). The control plate has unit arc length and a con-
+stant thickness roughly equal to 0.18 times the cylinder radius and is situated at
+a distance d from the cylinder surface. On the surface of the control plate, im-
+permeability and no-slip boundary conditions are considered. The control plate is
+kept constant at the same temperature as the free stream fluid.
+The nondimensional stream-function-vorticity formulation of the 2-D Navier-
+Stokes equations and energy equation in polar coordinates (r,θ) are given as,
+∂ 2ω
+∂r2 + 1
+r
+∂ω
+∂r + 1
+r2
+∂ 2ω
+∂θ2 = Re
+2
+�
+u∂ω
+∂r + v
+r
+∂ω
+∂θ + ∂ω
+∂t
+�
+(1)
+∂ 2ψ
+∂r2 + 1
+r
+∂ψ
+∂r + 1
+r2
+∂ 2ψ
+∂θ2 = −ω
+(2)
+∂ 2φ
+∂r2 + 1
+r
+∂φ
+∂r + 1
+r2
+∂ 2φ
+∂θ2 = RePr
+2
+�
+u∂φ
+∂r + v
+r
+∂φ
+∂θ + ∂φ
+∂t
+�
+(3)
+5
+
+Nomenclature
+Re
+Reynolds number (= 2R0U∞/ν)
+Pr
+Prandtl number (= ν/β)
+R0
+Radius of the circular cylinder
+R∞
+Radius of the far field boundary
+d
+Dimensional distance of the control plate
+from the cylinder surface
+U∞
+The free-stream fluid’s velocity
+T∞
+The free-stream fluid’s temperature
+ˆt, t
+Time in dimensional and nondimensional form
+Ts
+Surface temperature of the cylinder in dimensional form
+ˆα, α
+Rotational velocity in dimensional and
+nondimensional form (α = ˆαR0/U∞)
+d
+Distance of control plate from the surface of the cylinder
+Nu, Nu, Nut
+Nusselt number (local, average, and time-averaged total)
+h, havg
+Coefficients of heat transfer (local and average)
+ν
+The fluid’s kinematic viscosity
+K
+The fluid’s thermal conductivity
+β
+The fluid’s thermal diffusivity
+Q′′
+Radial heat flux on the surface (Local)
+ˆψ, ψ
+Stream function in dimensional and nondimensional form
+ˆω, ω
+Vorticity in dimensional and nondimensional form
+T, φ
+Temperature in dimensional and nondimensional form
+ˆu, u
+Radial velocity in dimensional and nondimensional form
+ˆv, v
+Tangential velocity in dimensional and nondimensional form
+ˆr, r
+Radius in dimensional and nondimensional form
+6
+
+Figure 1: The schematic illustration of the current problem.
+The velocities, v and u can be expressed as
+v = −∂ψ
+∂r
+and
+u = 1
+r
+∂ψ
+∂θ
+(4)
+ω can be written as
+ω = 1
+r
+� ∂
+∂r(vr)− ∂u
+∂θ
+�
+(5)
+The boundary conditions on the cylinder’s surface include impermeability, no-
+slip, and constant temperature, i.e.
+ψ = 0,
+∂ψ
+∂r = −α
+and
+φ = 1.0
+when
+r = 1
+(6)
+The condition of surface vorticity is provided by
+ω = −∂ 2ψ
+∂r2
+when
+r = 1
+(7)
+7
+
+Uoo
+Too
+()0
+u.(r,0) = U
+cos6
+d
+u(r,0,t) =0
+0=0
+v(r,0,t) = 0
+ControlPlate
+Ts
+u
+Uo
+V(r,0)
+sing
+r2
+Too
+V
+Uoo
+Roo
+Rco
+XIn the distant field, R∞, the vorticity’s resulting decay and the free-stream con-
+dition are taken to constitute the boundary conditions, i.e.
+ψ →
+�
+r − 1
+r
+�
+sin θ,
+∂ψ
+∂r →
+�
+1+ 1
+r2
+�
+sin θ
+and φ → 0 as r → R∞
+R0
+(8)
+ω → 0 as r → R∞
+R0
+(9)
+The criteria Eqs. (6) to (9) must be followed by all the parameters with 0 ≤
+θ ≤ 2π for all θ. In addition, all of the parameters are functions of θ with a period
+of 2π. The initial conditions for the stream function are given by Eqs. (8) and (9).
+The vorticity in the distant field is initially assumed to be zero. Eqs. (4) and (8)
+provide the initial requirements for the velocities as follows:
+u =
+�
+1− 1
+r2
+�
+cos θ
+and v = −
+�
+1+ 1
+r2
+�
+sin θ
+(10)
+3. Numerical Scheme
+Using a temporally second order accurate and spatially atleast third order accu-
+rate higher order compact (HOC) finite difference technique [32–35], the govern-
+ing equations of motion and the energy equation are discretized on non-uniform
+polar grids in the circular region ([R0,R∞] × [0,2π]) with grid points (ri,θj).
+The non-uniform grid concentrated around the cylinder is generated using the
+stretching function ri = exp
+�λπi
+imax
+�
+, 0 ≤ i ≤ imax. The function θj is given by,
+θj = 2π j
+jmax
+, 0 ≤ j ≤ jmax. The discretized equations can be written as [30, 36, 37]:
+[X1i jδ 2
+r +X2i jδ 2
+θ +X3i jδr +X4i jδrδθ +X5i jδrδ 2
+θ
++X6i jδ 2
+r δθ +X7i jδ 2
+r δ 2
+θ ]ψi j = Gi j
+(11)
+[Y11i jδ 2
+r +Y12i jδ 2
+θ +Y13i jδr +Y14i jδθ +Y15i jδrδθ
++Y16i jδrδ 2
+θ +Y17i jδ 2
+r δθ +Y18i jδ 2
+r δ 2
+θ ]ωn+1
+i j
+= [Y21i jδ 2
+r +Y22i jδ 2
+θ +Y23i jδr +Y24i jδθ +Y25i jδrδθ
++Y26i jδrδ 2
+θ +Y27i jδ 2
+r δθ +Y28i jδ 2
+r δ 2
+θ ]ωn
+i j
+(12)
+8
+
+and
+[Z11i jδ 2
+r +Z12i jδ 2
+θ +Z13i jδr +Z14i jδθ +Z15i jδrδθ
++Z16i jδrδ 2
+θ +Z17i jδ 2
+r δθ +Z18i jδ 2
+r δ 2
+θ ]φn+1
+i j
+= [Z21i jδ 2
+r +Z22i jδ 2
+θ +Z23i jδr +Z24i jδθ +Z25i jδrδθ
++Z26i jδrδ 2
+θ +Z27i jδ 2
+r δθ +Z28i jδ 2
+r δ 2
+θ ]φn
+i j
+(13)
+The coefficients X1i j, X2i j,..., X7i j; Gi j; Y11i j, Y12i j,..., Y18i j; Y21i j, Y22i j,...,
+Y28i j; Z11i j, Z12i j,..., Z18i j and Z21i j, Z22i j,..., Z28i j are the functions of the
+parameters r and θ. [30, 36, 37] provide the expressions for the non-uniform
+central difference operators δθ, δ 2
+θ , δr and δ 2
+r , as well as the notations θf , θb, r f, rb
+and the coefficients. The Bi-conjugate Gradient Stabilized approach is employed
+in order to solve the discretized problem.
+3.1. Drag and lift coefficients
+The forces acting on a circular cylinder submerged in fluids for uniform flow
+are generally caused by surface friction and surface pressure distribution. The
+expressions for drag (CD) and lift (CL) coefficients are adopted from [30, 36]. The
+expressions are as follows,
+CD = 1
+Re
+� 2π
+0
+��∂ω
+∂r
+�
+R0
+−ωR0
+�
+cosθdθ
+(14)
+CL = 1
+Re
+� 2π
+0
+��∂ω
+∂r
+�
+R0
+−ωR0
+�
+sinθdθ
+(15)
+The integral values are calculated using Simpson’s 1/3 method. The time-
+averaged drag, CD is expressed as,
+CD =
+1
+t1 −t2
+� t2
+t1
+CDdt
+(16)
+When the flow achieves a periodic mode and executes numerous cycles, the time
+span between t1 and t2 is selected.
+9
+
+3.2. The heat transfer parameters
+Initially, heat conduction happens from the cylinder surface to the adjacent
+fluid, and subsequently it convects away with the flow. The heat conduction path
+follows the radius of the cylinder surface. The dimensionless local heat flux in the
+radial direction is the local Nusselt number, Nu, defined by,
+Nu = 2hR0
+k
+= Q′′(2R0)
+k(Ts −T∞)
+(17)
+where h represents the local heat transfer coefficient, k represents the thermal
+conductivity of the fluid, and Q′′ represents the surface local radial heat flux. Q′′
+is expressed as, Q′′ = −k ∂T
+∂r |r=R0. The average Nusselt number, denoted by Nu,
+used to represent the dimensionless heat transfer from the cylinder’s surface, is
+expressed as
+Nu = 2havgR0
+k
+= 1
+2π
+� 2π
+0
+Nudθ
+(18)
+The average heat transfer coefficient (havg) is expressed as havg =
+1
+2π
+� 2π
+0 hdθ.
+Nut, the time-averaged total Nusselt number is given as,
+Nut =
+1
+t1 −t2
+� t2
+t1
+Nudt
+(19)
+When the flow achieves a periodic mode and executes numerous cycles, the time
+span between t1 and t2 is selected.
+3.3. Validation
+The computational domain is discretized using non-uniform grids. The grid
+independence test is performed in Fig. 2(a) with three different grid sizes (181 ×
+181), (191 ×202) and (351 ×341), with a set time step ∆t = 0.01, a fixed 25 : 1
+domain-to-cylinder-radius ratio, Pr = 0.7, Re = 150, α = 1 and d/R0 = 1. All the
+grid sizes seem to produce almost same results. The grid size (191×202) is cho-
+sen for future computations. For grid size (181 × 181) and time step ∆t = 0.01,
+the domain independence test is performed in Fig. 2(b) for three distinct radii,
+15, 25 and 35 of the outer boundary, other parameter values, on the other hand,
+are treated the same as the grid independence test. This test demonstrates that
+a domain radius of 25 is adequate to provide the best possible results. Finally,
+with a set grid size (181 ×181) and the far field border defined at 25 : 1 domain-
+to-cylinder-radius ratio, the time independence test is conducted in Fig. 2(c) for
+10
+
+(a)
+(b)
+(c)
+Figure 2: Variation of local Nusselt number distribution, Nu (a) grid independence test with grid
+sizes 181 × 181, 191 × 202, 351 × 341, (b) space independence test with outer boundary radius
+15, 25, 35 and (c) time independence test with time steps 0.001, 0.005, 0.01 at instant t = 10 for
+Re = 150, Pr = 0.7, α = 1 and d = 1.
+time increments ∆t = 0.001, 0.005, 0.01, 0.02. For later computations, we used
+R∞
+R0 = 25 and ∆t = 0.01, as suggested by these test findings.
+There hasn’t been any research towards controlling heat and flow transfer from
+a rotating cylinder using an arc-shaped vertical control plate placed across a free
+stream of uniform flow. To prove the correctness of our code and model, we be-
+gin by comparing our findings to those of previous studies of heat transfer from
+11
+
+15
+At=0.001
+At=0.005
+At=0.01
+12
+9
+nN
+6
+0
+60
+120
+180
+240
+300
+360
+015
+15
+25
+35
+10
+nN
+60
+120
+180
+240
+300
+360
+015
+181 X 181
+191 X 202
+8
+351 X 341
+10
+Nu
+5
+11
+1
+0
+60
+120
+180
+240
+300
+360
+0Table 1: Comparison of the current computation with the equivalent time-averaged total Nusselt
+number computed by Paramane & Sharma [22] for Re = 40, 100, Pr = 0.7, α = 1, and isother-
+mally heated cylinder.
+Re
+40
+100
+Nut (Current)
+3.276112
+4.936597
+Nut (Paramane & Sharma)
+3.213
+4.991
+Di f ference(%)
+1.964%
+1.09%
+Table 2: Comparison between time-averaged drag results from the current study to Kwon and
+Choi’s work [11] for Re = 160.
+Length of splitter plate
+1
+2
+Time-averaged Drag (Present Study)
+1.133021
+1.056131
+Time-averaged Drag (Kwon and Choi)
+1.10162
+1.08812
+Di f ference(%)
+2.85%
+2.94%
+rotating cylinders [22], then it is compared with the results of flow past circular
+cylinder with a splitter plate attached [11]. When the flow becomes periodic, the
+mean drag coefficients are used to determine the time-averaged drag coefficient
+on the cylinder surface in Table 2. According to Table 1, the maximum difference
+of time-averaged total Nusselt number is 1.964%, which is within a reasonable
+range. Also, Table 2 shows the maximum difference of time-averaged Drag coef-
+ficients from the results of the current study and the previously published works is
+2.94% which is also within a considerable range. As a result, the current findings
+are consistent with earlier studies.
+4. Results and Discussions
+Reynolds number (Re), Prandtl number (Pr), angular velocity of the cylinder
+(α), and control plate distance (d/R0) are all well-known factors that influence
+flow and heat fields. Fig. 3 exhibits the Drag (CD) and lift (CL) coefficients, as
+well as the variation of local Nusselt number (Nu) for d/R0 = 0 and 0.5 with
+fixed α = 0.5, Re = 150, Pr = 0.7. The parameter value d/R0 = 0 corresponds
+to the case without the arc-shaped control plate. Fig. 3(a) clearly demonstrates
+that, the peak value of CD is significantly reduced as well as the amplitude of
+CL with the introduction of control plate at a distance d/R0 = 0.5 downstream.
+d/R0 = 0, indicating flow across the cylinder in the absence of the control plate.
+By comparing Fig. 3(b) and Fig. 3(c), it is found that the introduction of the con-
+12
+
+(a)
+(b)
+(c)
+Figure 3: (a) Drag (CD) and lift (CL) coefficients, (b) variation of local Nusselt number (Nu) for
+d/R0 = 0 and (c) variation of local Nusselt number (Nu) for d/R0 = 0.5 with fixed α = 0.5.
+trol plate slightly reduced the peak value of Nu approximately from 11.35 to 11.27
+at θ ≈ 192◦, but the local maximum peak is significantly increased at θ ≈ 30◦.
+It means, although the heat transfer near the front stagnation is slightly decreased
+by the control plate, the heat transfer is significantly increased near the rear stag-
+nation point, which eventually increases the overall heat transfer from the upper
+half of the cylinder surface. The control plate alters the vortex shedding process,
+which in turn affects the thermal boundary layer and causes this effect. Realizing
+the importance of the arc-shaped control plate, the current studies are performed
+for Re = 150, 0.5 ≤ α ≤ 3.25 and 0.5 ≤ d/R0 ≤ 3, while Pr is maintained at 0.7.
+The values of α are typically chosen in accordance with [31].
+For α = 0.5 and d/R0 = 1, Fig. 4 exhibits the isotherm, streamline and vor-
+ticity at periodic phases. Two vortices are shed periodically from the upper and
+lower sides of the cylinder, according to the vorticity and streamline. The upper
+vortex is slightly larger than the lower vortex. The continuous and dashed lines
+indicate the positive and negative contours, respectively. The vortex shedding
+plane is shifted by approximately θ = 20◦ from the centerline or the x-axis due
+to the rotation of the cylinder. The shear layer around the plate changes the nega-
+13
+
+16.5
+11.27
+t = t+(O)T
+15
+t = t +(1/4)T
+13.5
+1.265
+t = t,+(1/2)T
+t = t,+(3/4)T
+12
+11.26
+190
+192
+19.
+t = t,+(1)T
+10.5
+7.5
+6
+4.5
+3
+1.5
+60
+120
+240
+300
+36016.5
+t = t+(O)T
+15
+11.35
+t = t +(1/4)T
+13.5
+t = t,+(1/2)T
+t = t,+(3/4)T
+12
+11.3
+190
+195
+t = t,+(1)T
+10.5
+7.5
+6
+4.5
+3
+1.5
+60
+120
+180
+240
+300
+360
+03
+Cp, d/R, = 0
+Cp, d/R, = 0.5
+CL, d/R.
+=0
+d/R.
+= 0.5
+0
+50
+100
+150
+tt = t0 +(0)T
+t = t0 +(1/4)T
+t = t0 +(1/2)T
+t = t0 +(3/4)T
+t = t0 +(1)T
+(a)
+(b)
+(c)
+Figure 4: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 0.5
+and d/R0 = 1 at different phases.
+tive equi-vorticity lines that come from the surface of the cylinder, but the positive
+equi-vorticity lines from the cylinder merge with the shear layer due to the rotation
+of the cylinder. No recirculation zone or vortex is observed between the cylinder
+and the plate. Two large vortices as lumps of hot fluid shed periodically from the
+upper and bottom sides of the cylinder according to the isotherm contours. The
+14
+
+**
+!
+.......:2
+:i..
+:
+2t = t0 +(0)T
+t = t0 +(1/4)T
+t = t0 +(1/2)T
+t = t0 +(3/4)T
+t = t0 +(1)T
+(a)
+(b)
+(c)
+Figure 5: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 1 and
+d/R0 = 1 at different phases.
+isotherm density is high near the front stagnation point, which indicates the higher
+heat transfer rate in this region. Isotherm, streakline and vorticity are displayed
+in Fig. 5 for α = 1 and d/R0 = 1. Streakline and vorticity indicate that two vor-
+tices are periodically shed from the upper side and lower side of the cylinder. The
+increase in rotational rate, increases the movement of the fluid around the control
+15
+
+:
+.....t = t0 +(0)T
+t = t0 +(1/4)T
+t = t0 +(1/2)T
+t = t0 +(3/4)T
+t = t0 +(1)T
+(a)
+(b)
+(c)
+Figure 6: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 2.07
+and d/R0 = 1 at different phases.
+plate which leads to thickening of shear layer around the control plate. It affects
+the vorticity contour coming from the cylinder. Positive equi-vorticity lines from
+the cylinder and the plate get merged together to shed a sleek, elongated vortex.
+The positive equi-vorticity lines from the cylinder completely cover the control
+plate, also dragging the negative equi-vorticity lines towards the bottom of the
+16
+
+t = t0 +(0)T
+t = t0 +(1/4)T
+t = t0 +(1/2)T
+t = t0 +(3/4)T
+t = t0 +(1)T
+(a)
+(b)
+(c)
+Figure 7: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 3.25
+and d/R0 = 1 at different phases.
+cylinder. This increases the density in thermal boundary layein the upper half of
+the cylinder, increasing the heat transfer. The upper vortex is much wider as com-
+pared to the sleek bottom vortex. Because of the increased α, the vortex shedding
+plane is shifted by approximately θ = 23◦ from the centerline. The isotherm con-
+tours suggest that two warm blobs convect away periodically from the upper and
+17
+
+lower sides of the cylinder. There is no vortex or recirculation zone found be-
+tween the cylinder and the plate. Isotherm, streakline and vorticity for α = 2.07
+and d/R0 = 1 are shown in Fig. 6. The streakline and vorticity suggest that two
+vortices are periodically shed in the flow domain. One vortex is shed from the top
+of the cylinder and the other one is shed from the back of the plate. The lower
+vortex pushes the upper vortex due to the high rotation rate of the cylinder. As a
+result, the upper vortex is shed much earlier than at lower rotational rates. Also,
+the upper vortex becomes sleek and the bottom vortex becomes wide. Negative
+equi-vorticity lines coming from the cylinder completely cover the control plate
+as well as the positive vortex. Due to the high movement of fluid around the con-
+trol plate, the shear layers get thickened and drag the negative equi-vorticity lines
+from the cylinder to the bottom of the plate. This affects the thermal boundary
+layer of the cylinder by thinning around the rear stagnation point. As a result, the
+heat transfer is increased in this region. However, the high rotation of the cylin-
+der thickens the thermal boundary layer around the front stagnation point, leading
+to a decrease in heat transfer rate. Here, The vortex shedding plane is shifted
+by approximately θ = 37◦ from the centerline. The isotherm contours suggest
+that two warm blobs convect away periodically by the vortices generated in the
+flow domain. Fig. 7 shows the isotherm, streakline and vorticity for α = 3.25
+and d/R0 = 1. Due to very high rotational rate, the negative equi-vorticity lines
+completely cover the cylinder as well as the positive equi-vorticity lines originated
+from the control plate. Two vortices are shed periodically, one from the top of the
+cylinder and another from the back of the control plate. Due to the high rotational
+speed, the bottom vortex pushes the upper vortex. As a result, the upper vortex is
+shed much earlier. Also, the bottom vortex is much larger than the upper vortex.
+One small negative vortex is formed behind the control plate, but it gets dissolved
+into the positive vortex. After the negative vortex is shed, the shear layer from
+the cylinder splits on top and bottom of the shear layer from the control plate. It
+gradually merges and creates an elongated negative vortex. Between the cylinder
+and the plate, no vortex or recirculation zone forms. The moving fluid around the
+cylinder drags the shear layer from the control plate towards the top of the cylin-
+der, which leads to the increased density of the isotherm contour. As a result, heat
+transfer is boosted in this region. The vortex shedding plane is displaced from the
+centerline by approximately θ = 50◦ at this rotational rate. The isotherm contours
+suggest that the density of the isotherm around the cylinder becomes less than at
+the lower rotational rates, which means that the high rotation rate is suppressing
+the heat transfer rate from the cylinder surface. Additionally, two warm blobs pe-
+riodically convect away from the cylinder’s upper side and the plate’s rear. The
+18
+
+t = t0 +(0)T
+t = t0 +(1/4)T
+t = t0 +(1/2)T
+t = t0 +(3/4)T
+t = t0 +(1)T
+(a)
+(b)
+(c)
+Figure 8: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 0.5
+and d/R0 = 2 at different phases.
+top blob is sleek and the bottom one is wide, similar to the vortices. Figs. 4 to 7
+show that increasing rotational rates increased the size of vortices as well as the
+angle of vortex shedding plane from the centerline for a fixed d/R0 = 1.
+Fig. 8 shows the isotherm, streakline and vorticity for α = 0.5 and d/R0 = 2.
+19
+
+::
+:t = t0 +(0)T
+t = t0 +(1/4)T
+t = t0 +(1/2)T
+t = t0 +(3/4)T
+t = t0 +(1)T
+(a)
+(b)
+(c)
+Figure 9: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 3.25
+and d/R0 = 2 at different phases.
+Two vortices are shed periodically from the upper and lower sides of the cylinder,
+according to the streakline and vorticity. One recirculation zone is formed by the
+interaction of the shear layers between the cylinder and the plate, near the top of
+the plate. Positive equi-vorticity lines originated from the cylinder partially covers
+the control plate. The isotherm contours show that two warm blobs convect away
+20
+
+with the shedding vortices. The vortex shedding plane is slightly higher than the
+centerline by approximately θ = 15◦. This angle of the vortex shedding plane
+is slightly lower than that of Fig. 4 due to the increase in d/R0. This happens
+due to the interaction of shear layers around the control plate. Fig. 9 exhibits the
+isotherm, streakline and vorticity for α = 3.25 and d/R0 = 2. Here, two vortices
+are periodically shed. One is shed from the top of the cylinder, and the other one
+is shed from behind the plate. One temporary recirculation zone is formed be-
+tween the cylinder and the plate, which gradually merges with the upper vortex.
+Due to the high rotational rate, the bottom vortex is pulled upwards and pushes
+the upper vortex. As a result, the upper vortex is shed much earlier. The negative
+equi-vorticity lines cover the positive equi-vorticity lines that originated from the
+control plate. After the negative vortex is shed, the shear layer is split into two
+by the positive vorticity contour. The shear layers from the top and bottom of the
+control plate are squeezed together by the negative vorticity contour to form the
+positive vortex. The vortex shedding plane is shifted by approximately θ = 40◦
+from the centerline, and this angle is also slightly lower than that of Fig. 7. The
+widths of vortices are much larger than those at Fig. 7. The interaction between
+the shear layer and the boundary layer of the cylinder thickens the thermal bound-
+ary layer near the front stagnation point and increases the density of the isotherm
+contour near the rear stagnation point and at the bottom of the cylinder. It leads
+to the reduction of heat transfer near the front stagnation point and an increase in
+heat transfer rate near the rear stagnation point and bottom of the cylinder. Figs. 8
+and 9 show that as α increases from 0.5 to 3.25 for d/R0 = 2, the vortices increase
+in size.
+Isotherm, streakline and vorticity are displayed in Fig. 10 for α = 0.5 and
+d/R0 = 3. Two vortices shed periodically from the upper and lower sides of the
+cylinder. The bottom vortex is slightly sleeker than the upper one. The positive
+equi-vorticity lines coming from the cylinder, partially cover the control plate,
+and the interaction between the shear layers sheds the positive vortex. One re-
+circulation zone is formed between the cylinder and the plate, which gradually
+merges with the upper vortex. The density of the isotherm contour is higher near
+the front stagnation point, which means the rate of heat transfer is much higher in
+this region. Also, two warm blobs convect away periodically from the upper and
+lower sides of the cylinder. Also, the vortex shedding plane is at an angle of ap-
+proximately θ = 8.5◦ with the centerline, which is much lower than the previous
+placements of the control plate. It happens as the bottom shear layers are resisted
+by the control plate to freely move upwards. Fig. 11 exhibits the isotherm, streak-
+21
+
+t = t0 +(0)T
+t = t0 +(1/4)T
+t = t0 +(1/2)T
+t = t0 +(3/4)T
+t = t0 +(1)T
+(a)
+(b)
+(c)
+Figure 10: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 0.5
+and d/R0 = 3 at different phases.
+line and vorticity are displayed for α = 3.25 and d/R0 = 3. Two vortices shed
+periodically behind the control plate. The rotational motion of the fluid surround-
+ing the cylinder causes the negative equi-vorticity lines to surround the cylinder
+as well as the positive equi-vorticity lines that originate from the control plate.
+The positive equi-vorticity lines also cover the control plate. The shear layers that
+22
+
+sitt:t = t0 +(0)T
+t = t0 +(1/4)T
+t = t0 +(1/2)T
+t = t0 +(3/4)T
+t = t0 +(1)T
+(a)
+(b)
+(c)
+Figure 11: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 3.25
+and d/R0 = 3 at different phases.
+originate from the cylinder get split after interaction with the shear layer around
+the control plate, and they merge together during the shedding of the negative vor-
+tex. Most of the fluid particles that flow across the cylinder are sucked down and
+flow below the control plate. This complex flow dynamics is the combined effect
+of the high rotational rate and the placement of the control plate. It reduces the
+23
+
+angle of the vortex shedding plane with the centerline to approximately θ = 12◦,
+which is much less than the previous placements of the control plate with this
+high rotational rate. Also, the size of the negative vortex is drastically reduced
+and becomes extremely sleek due to the interaction of shear layers. The lower
+vortex grows from the bottom of the plate and moves upwards. The density of the
+isotherm contour around the cylinder is very low due to the high rotational rate. As
+a result, the boundary layer thickens around the cylinder and suppresses the rate
+of force convective heat transfer. The isotherm contours indicate that two warm
+blobs periodically convect away from the upper side of the cylinder and the lower
+end of the control plate. Therefore, the placement of the control plate, together
+with the rotational rate, considerably suppressed the vortex shedding process as
+well as the heat convection. Figs. 10 and 11 illustrate that the size of the vortices
+grow as α increases from 0.5 to 3.25 for d/R0 = 3. Also, Figs. 4, 8 and 10 show
+that the wake length of vortices increases with increasing distance of the control
+plate from the cylinder surface at α = 0.5. It is also observed that the increasing
+distance of the control plate significantly decreases the angle of the vortex shed-
+ding plane with the centerline for respecting rotational rates.
+The drag (CD) and lift (CL) coefficients at different α with varying d/R0 are
+shown in Fig. 12. The figures show that the drag and lift coefficients are periodic
+in nature. For α = 0.5, the drag coefficient gradually decreases with increasing
+d/R0 and the lift coefficient is minimum at d/R0 = 0.5. The differences in the
+drag coefficients for this α = 0.5 are very small. There is not much difference
+in lift coefficient for d/R0 = 1, 2, and 3. When α = 1, gradual decrease in the
+drag coefficient is found with increasing d/R0. The lift coefficient is found to be
+minimum for d/R0 = 0.5. The maximum value of the lift coefficient is observed
+for d/R0 = 1 and 2. When α = 2.07, maximum value of the drag coefficient is
+found for d/R0 = 3 and minimum value is found for d/R0 = 0.5. Here, the maxi-
+mum value of the lift coefficient is found for d/R0 = 1 and the minimum value is
+found for d/R0 = 0.5. When the rotation rate is at its maximum, i.e., α = 3.25,
+the amplitudes of the drag and lift coefficients increase drastically for all d/R0.
+Here, the minimum values of lift and drag coefficients are found for d/R0 = 0.5
+and the minimum values are found d/R0 = 2. For d/R0 = 3, the amplitudes of the
+drag and lift coefficients are the smallest. So, the impact of various positionings
+of the arc-shaped control plate is significant at higher rotational rates. In Fig. 13,
+the drag (CD) and lift (CL) coefficients at different d/R0 with varying α are shown.
+At, d/R0 = 0.5, the maximum value of the CD is found for α = 1 and the mini-
+mum value is found for α = 3.25. With increasing α, the maximum value of CL
+24
+
+α = 0.5
+α = 1
+α = 2.07
+α = 3.25
+(a)
+(b)
+Figure 12: (a) Drag coefficient CD and (b) lift coefficient CL with varying d/R0.
+gradually decreases while the amplitude of CL gradually increases. The highest
+amplitude of the drag and lift coefficients is observed for α = 3.25. When the
+25
+
+4
+d/R. = 0.5
+d/R。= 1
+2
+d/R。= 2
+d/R. = 3
+0
+-6
+008
+220
+240
+260
+280
+300
+t5
+d/R. = 0.5
+d/R。= 1
+d/R。= 2
+d/R. = 3
+3
+2
+200
+220
+240
+260
+280
+300
+t4
+d/R. = 0.5
+d/R。= 1
+2
+d/R。= 2
+d/R. = 3
+0
+6
+003
+220
+240
+260
+280
+300
+t5
+d/R. = 0.5
+d/R。= 1
+d/R。= 2
+d/R.= 3
+3
+2
+220
+240
+260
+280
+300
+t4
+d/R. = 0.5
+d/R。= 1
+2
+d/R。= 2
+d/R.= 3
+0
+-1.2
+-1.4
+-1.6
+-6
+-1.8
+260
+280
+300
+003
+220
+240
+260
+280
+300
+t5
+d/R. = 0.5
+d/R。= 1
+d/R。= 2
+4
+d/R. = 3
+1.3
+3
+D
+1.2
+2
+250
+260
+270
+280
+290
+220
+240
+260
+280
+300
+t4
+d/R. = 0.5
+d/R。= 1
+2
+d/R。= 2
+d/R. = 3
+0
+-0.7
+-4
+-0.8
+-6
+-0.9
+260
+280
+300
+800
+220
+240
+260
+280
+300
+t5
+d/R. = 0.5
+d/R。= 1
+d/R。= 2
+4
+d/R. = 3
+3
+1.2
+D
+1.15
+C
+2
+1.1
+250
+260
+270
+280
+290
+220
+240
+260
+280
+300
+td/R0 = 0.5
+d/R0 = 1
+d/R0 = 2
+d/R0 = 3
+(a)
+(b)
+Figure 13: (a) Drag coefficient CD and (b) lift coefficient CL with varying α.
+26
+
+4
+α = 0.5
+α=1
+2
+α = 2.07
+α = 3.25
+0
+4
+-6
+003
+220
+240
+260
+280
+300
+tin
+α = 0.5
+α=1
+4
+α = 2.07
+α = 3.25
+3
+D
+2
+200
+220
+240
+260
+280
+300
+t4
+α = 0.5
+α=1
+2
+α = 2.07
+α = 3.25
+0
+-4
+-6
+003
+220
+240
+260
+280
+300
+t5
+α = 0.5
+α=1
+4
+α = 2.07
+α = 3.25
+3
+D
+2
+220
+240
+260
+280
+300
+t4
+α = 0.5
+α=1
+2
+α = 2.07
+α = 3.25
+0
+A
+-6
+800
+220
+240
+260
+280
+300
+t5
+α = 0.5
+α=1
+A
+α = 2.07
+α = 3.25
+3
+2
+220
+240
+260
+280
+300
+t4
+α = 0.5
+α=1
+2
+α = 2.07
+α = 3.25
+0
+008
+220
+240
+260
+280
+300
+t5
+α = 0.5
+α=1
+A
+α = 2.07
+α = 3.25
+3
+2
+220
+240
+260
+280
+300
+tθ
+θ
+θ
+(a)
+(b)
+(c)
+θ
+θ
+θ
+(d)
+(e)
+(f)
+θ
+θ
+(g)
+(h)
+Figure 14: Local Nusselt number variation at periodic phases for (a) d/R0 = 1, α = 0.5; (b)
+d/R0 = 1, α = 1; (c) d/R0 = 1, α = 2.07; (d) d/R0 = 1, α = 3.25; (e) d/R0 = 2, α = 0.5; (f)
+d/R0 = 2, α = 3.25; (g) d/R0 = 3, α = 0.5; and (h) d/R0 = 3, α = 3.25.
+plate distance is increased to 1 and 2, the maximum value of CD and the minimum
+value of CL are found for α = 3.25. Also, the amplitudes are maximum for the
+highest rotational rate. When d/R0 = 3, CD gradually increases while CL gradu-
+ally deceases as α increases. The lift coefficients suggest that the lock-on vortices
+are shed under all the considered rotational rates and distances of the control plate.
+Fig. 14 shows the variation of local Nusselt numbers at periodic phases for var-
+27
+
+ious rotational rates of the cylinder and different positioning of the plate. Fig. 14(a)
+shows the variation of Nu for d/R0 = 1 and α = 0.5. It can be seen that the
+maximum value of Nu is slightly shifted downwards from the front stagnation
+point (θ = 180◦) approximately to θ = 192◦. It indicates the difference in heat
+transfer processes between the upper and lower half of the cylinder surface. The
+differences in values of Nu between the periodic phases are very small. A local
+maximum peak of Nu is found at θ ≈ 30◦ which indicates the higher rate of heat
+convection in this area. This is supported by the concentrated isotherm contours
+in this area close to the cylinder surface shown in Fig. 4. As the α increases to
+2 for d/R0 = 1 in Fig. 14(b), the differences in values are increased at different
+phases. The highest point of Nu is found around θ = 204◦. It shows the difference
+in heat transfer mechanisms from the upper and lower surfaces. A local maximum
+peak of Nu is found at θ ≈ 42◦ indicating higher rate of heat convection in this
+area. It is also supported by the highly concentrated isotherm contours in Fig. 5.
+When α = 2.07 for d/R0 = 1, the maximum value of Nu slightly decreases in
+Fig. 14(c) than that the previous cases and the maximum point of heat transfer is
+around θ = 240◦. Local maximum peak is found to be changing position between
+θ ≈ 42◦ and θ ≈ 78◦ at different periodic phases due to the complex vortex shed-
+ding phenomenon. These areas convect a large amount of heat into the fluid. The
+asymmetric Nu-distribution around the front stagnation point shows that the heat
+transfer process from the upper part of the cylinder surface is far different from
+the heat transfer process from the lower part of the cylinder surface. Fig. 14(d)
+shows the variation of Nu with maximum rotation rate, α = 3.25 for d/R0 = 1 and
+the maximum value of Nu drastically decreases and occurs at θ ≈ 72◦ i.e. near
+the rear stagnation point. As many researchers previously mentioned, here too,
+large rotational rates significantly reduce the maximum heat transfer rate from the
+cylinder [19, 22, 23]. A local maximum peak is found at θ ≈ 252◦. The reduc-
+tion of the maximum peak value of Nu at front stagnation point with increasing α
+hints to the fact that more heat is transferred under conduction in this area. This
+happens due to the thickening of the boundary layer around the cylinder surface
+at the high rotational rate. Fig. 14(e) shows the variation of Nu with α = 0.5 and
+d/R0 = 2. It shows that the maximum point of heat transfer is around θ = 191◦.
+Also, the peak value is slightly lower than that of d/R0 = 1 due to the vortex shed-
+ding process. A local maximum peak is found at θ ≈ 24◦ i.e. the heat transfer is
+higher in this area. It is also supported by the respective dense isotherm contours.
+In Fig. 14(f), α is increased to 3.25 for d/R0 = 2 and it is found that the highest
+value of Nu significantly reduced than the previous case. The highest value of Nu
+is observed around θ = 264◦ i.e. maximum heat transfer under convection occurs
+28
+
+Nut
+α
+α
+α
+α
+α
+Nut
+(a)
+(b)
+Figure 15: (a) Nut for varying α, and (b) Nut for varying d/R0.
+in this area. This is supported by the respective dense isotherm contour in this
+region close to the cylinder surface. A local maximum value of Nu is found at
+θ ≈ 60◦ at periodic phases t = t0 + (0)T, t0 + (1)T, i.e. the heat transfer is en-
+hanced in this area under convection by the complex vortex shedding process. The
+Nu-distribution at 180◦ ≤ θ ≤ 0◦ is significantly different than the Nu-distribution
+at 360◦ ≤ θ ≤ 180◦. This demonstrates that the lower half of the cylinder surface
+convects more heat than the upper half. Fig. 14(g) shows the variation of Nu for
+α = 0.5 and d/R0 = 3. The highest value of Nu is observed around θ = 192◦.
+The maximum value is slightly lower than that of d/R0 = 1, 2. A local maximum
+value of Nu distribution is found at θ ≈ 24◦. The maximum heat transfer under
+convection occurs in these areas. The highest value of Nu-distribution curve is
+significantly reduced in Fig. 14(h) where α = 3.25 and d/R0 = 3 as compared to
+Fig. 14(d) for d/R0 = 1 and Fig. 14(f) for d/R0 = 2. This indicates that the in-
+creasing distance of the control plate significantly reduced the heat transfer under
+convection for the fixed α. The highest value of Nu is shifted to θ ≈ 276◦ due
+to the complex vortex shedding. The lowest value of Nu-distribution curve at the
+front stagnation point indicates that a large amount of heat is transferred by con-
+duction at this place. Also, the distribution curve at 180◦ ≤ θ ≤ 0◦ is significantly
+different than the curve at 360◦ ≤ θ ≤ 180◦, which shows that the lower half of
+the cylinder surface convects more heat than the upper half.
+Fig. 15 exhibits the variation of time-averaged total Nusselt number (Nut) with
+Fig. 15(a) varying α and Fig. 15(b) varying d/R0. The values of Nut for α = 0.5
+29
+
+are 6.672265, 6.251865, 6.154835 and 6.074185 with d/R0 = 0.5, 1, 2 and 3
+respectively. It means that the increasing distance of control plate reduces the
+heat transfer rate at α = 0.5. The values of Nut for α = 1 are 6.790804, 6.28877,
+6.07076 and 5.89388 with d/R0 = 0.5, 1, 2 and 3 respectively. It means that the
+increasing distance of control plate also reduces the heat transfer rate at α = 1.
+The values of Nut for α = 2.07 are 6.686615, 5.774501, 5.6436 and 5.643545
+with d/R0 = 0.5, 1, 2 and 3 respectively. Here also, the increasing distance of
+control plate reduces the heat transfer rate. The values of Nut for α = 3.25 are
+6.68507, 5.899766, 4.931795 and 4.9757 with d/R0 = 0.5, 1, 2 and 3 respectively.
+Again the increasing distance of the control plate reduces the heat transfer rate
+except for d/R0 = 3. This occurs due to the interaction of high rotation and the
+large distance of the control plate. Fig. 15(a) shows that Nut gradually deceases
+with increasing α at d/R0 = 2, 3 and the maximum value of Nut is found for
+d/R0 = 0.5, α = 0.5. Fig. 15(b) shows that increasing d/R0 significantly reduces
+Nut within the range of 0.5 ≤ d/R0 ≤ 2 for all rotational rates. However, if we
+place the plate further at a distance d/R0 = 3, not much change occurs. It is
+found from the comparison of maximum and minimum values of Nut that certain
+positioning of the control plate and rotational rate can enhance the heat transfer
+rate by 37.69%.
+5. Conclusion
+We numerically examined the control of a uniform, viscous fluid flow past
+circular cylinder by an arc-shaped plate positioned in the normal direction behind
+an isothermally heated circular cylinder rotating in the cross stream. The gov-
+erning equations are discretized using a HOC finite difference technique, and the
+system of algebraic equations obtained by the HOC discretization is solved using
+the Bi-conjugate gradient stabilised iterative method. According to the research,
+the distance between the control plate and the cylinder surface has a considerable
+impact on fluid flow along with the rotation of the cylinder. The structure of the
+wake changes depending on the position of the plate. When α is less than 1 with
+d/R0 = 1, two vortices as lumps of hot fluid are shed periodically from either
+side of the cylinder; when α is greater than 2.07 with d/R0 = 1, a large nega-
+tive vortex of heated fluid is shed from the upper side of the cylinder and another
+positive vortex of hot fluid is shed behind the control plate on a periodic basis.
+The increasing rotational rates increase the size of vortices and decrease the wake
+length for all positions of the control plate. The vortex shedding plane is shifted
+from the centerline by the cylinder’s rotational motion. For all rotational rates, the
+30
+
+increased distance of the control plate decreases the angle of the vortex shedding
+plane with the centerline, but the angle is increased with increasing rotational rates
+for all positions of the control plate. At higher rotational rates, the positive vortex
+is pulled upwards due to the interaction of fluid, and it pushes the negative vortex,
+causing an early shedding of it. Placing the control plate at d/R0 = 3 along with
+a high rotational rate is found to significantly reduce the size of vortices. It is also
+found that the impact of various positionings of the arc-shaped control plate is
+significant at higher rotational rates. An additional recirculation zone is found for
+(d/R0 = 2, α = 0.5, 3.25) and (d/R0 = 3, α = 3.25). Drag and lift coefficients
+for all 0.5 ≤ d/R0 ≤ 3 and 0.5 ≤ α ≤ 3.25 have a periodic nature . The values of
+drag and lift coefficients can be reduced or increased by utilising the rotation of the
+cylinder and the placement of the plate. The maximum value of drag coefficient
+is achieved for d/R0 = 2 and α = 3.25 which is about 3. All vortices shed are
+locked-on under the scope of considered parameters. It is found that the rotational
+rates relocate the highest point of heat transfer further from the front stagnation
+point, i.e., increasing the heat transfer by conduction in this region. The increasing
+distance of the control plate significantly reduced the heat transfer under convec-
+tion for the fixed α. The combined effect of rotation and the positioning of the
+control plate causes a different heat transfer mechanism at the upper half of the
+cylinder surface than at the lower half. For fixed d/R0 = 2 and α = 3.25, the
+maximum point of heat transfer is shifted towards the rear stagnation point from
+the front stagnation point due to the complex vortex shedding.
+Author Declarations
+The authors have no conflicts to disclose.
+Data Availability Statement
+The data that support the findings of this study are available from the corre-
+sponding author upon reasonable request.
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+arXiv:2301.11947v1  [math.DS]  27 Jan 2023
+A survey on Lyapunov functions for epidemic
+compartmental models
+N. Cangiotti∗, M. Capolli†, M. Sensi‡, S. Sottile§
+Abstract
+In this survey, we propose an overview on Lyapunov functions for a variety of com-
+partmental models in epidemiology. We exhibit the most widely employed functions,
+together with a commentary on their use.
+Our aim is to provide a comprehensive
+starting point to readers who are attempting to prove global stability of systems of
+ODEs. The focus is on mathematical epidemiology, however some of the functions and
+strategies presented in this paper can be adapted to a wider variety of models, such as
+prey-predator or rumor spreading.
+Mathematics Subject Classification:
+34D20, 34D23, 37N25, 92D30.
+Keywords:
+Epidemic models, Lyapunov functions, Compartmental models, Global
+stability, Ordinary Differential Equations, Disease Free and Endemic Equilibria.
+1
+Introduction
+Stemming from the pioneering work of Kermack and McKendrick [30], the mathematical
+modelling of infectious diseases has developed, over the last century, in various directions.
+An abundance of approaches and mathematical techniques have been employed to capture
+the many facets and details which describe the spread of an infectious disease in a population.
+In particular, compartmental models remain one of the most widely employed approaches.
+In these models, a population is partitioned into compartments, characterizing each individ-
+ual with respect to its current state in the epidemic.
+One can then write a system of
+Ordinary Differential Equations (from here onwards, ODEs) to study the evolution in time
+of the disease.
+∗Politecnico di Milano, Department of Mathematics, via Bonardi 9, Campus Leonardo, 20133, Milan
+(Italy). E-mail: [email protected]
+†Institute of Mathematics, Polish Academy of Sciences, Jana i Jedrzeja Sniadeckich 8, 00-656, Warsaw,
+(Poland). E-mail: [email protected]
+‡MathNeuro Team, Inria at Université Côte d’Azur, 2004 Rte des Lucioles, 06410, Biot, (France). E-mail:
+[email protected]
+§Department of Mathematics, University of Trento, Via Sommarive 14, 38123, Povo, (Italy). E-mail:
+[email protected]
+1
+
+These models usually take their names from the compartments they consider, the most
+renowned one being the Susceptible-Infected-Recovered (SIR) model. The SIR models can
+be extended to SIRS models by considering the acquired immunity to be temporary rather
+than permanent, allowing Recovered individuals to become Susceptible again. Various com-
+partments can be added, depending on the characteristic of the specific disease under study:
+Asymptomatic, Exposed, Waning immunity and many others.
+A remarkably useful tool for the study of this kind of models are Lyapunov functions,
+which ensure global (or, in some cases, local) asymptotic convergence towards one of the
+equilibria of the system.
+Given a system of n ODEs X′ = f(X) and an equilibrium point X∗, we call a scalar
+function V ∈ C1(Rn, R) a Lyapunov function if the following hold:
+1. V attains its minimum at X = X∗;
+2. V ′ = ∇V · f < 0 for X ̸= X∗.
+The classical definition of Lyapunov function requires also the conditions
+3. X∗ = 0 and V (X∗) = 0;
+however, these amount to a change of coordinates in Rn and a vertical translation of V , so
+we will accept the more general definition. The existence of such a function guarantess the
+global stability of the equilibrium X∗, as orbits of the systems naturally evolve towards the
+minimum power level of V .
+The Basic Reproduction Number R0 is a well know threshold in epidemics models. Usu-
+ally, R0 < 1 suggests Global Asymptotic Stability (from here onwards, GAS) of the Disease
+Free Equilibrium (from here onwards, DFE), whereas R0 > 1 suggests GAS of the Endemic
+Equilibrium (from here onwards, EE). In more complex models, the aforementioned condi-
+tions on R0 might not be sufficient to prove the GAS of either equilibria, especially in cases
+in which the EE is not unique. Lyapunov functions often explicitly involve R0 to guarantee
+the extinction of the disease or its endemicity over time.
+Unfortunately, given a generic system of ODEs, there is no universal way of deriving a
+Lyapunov function, nor to rule out the existence of one. However, there exist a few Lyapunov
+functions which have proven quite effective in a variety of different models.
+In this survey, we collect some of the most relevant functions available in the literature, to
+provide the reader with a series of options to apply to the model of their interest, depending
+on its formulation. We include an extensive bibliography to complement the essential infor-
+mation of each model we present. This will provide the reader with a convenient starting
+point to investigate the availability of a known Lyapunov function to analytically prove the
+asymptotic behaviour of their system of ODEs. For the sake of brevity, we do not repeat
+the proofs to show that any of the functions we present are, indeed, Lyapunov function
+for the respective system of ODEs. These proofs can be found in the papers we cite when
+introducing each model.
+Consider a model with compartments X1, X2, . . . , Xn. Then, the DFE has coordinates
+Xi = 0 for all i ∈ I, where I is the set of the indexes of infectious compartments, and the
+EE, which we indicate with (X∗
+1, X∗
+2, . . . , X∗
+n), has all positive entries. A vast majority of
+Lyapunov functions in epidemic modelling fall into one of the categories listed below.
+2
+
+1. Linear combination of infectious compartments. The Lyapunov function for the
+DFE when R0 < 1 is of the form
+L =
+�
+i≥2
+ciXi,
+for some constants ci ≥ 0 to be determined [6, 16, 18, 21, 32, 36, 39, 45, 49, 50, 59, 64,
+70]. To prove convergence of the system to the DFE in this case it is often required the
+use of additional tools, such as LaSalle’s invariance principle, which we briefly recall
+at the end of Section 2.1.
+2. Goh-Lotka-Volterra. The Lyapunov function for the EE when R0 > 1 is of the form
+L =
+�
+i
+ci(Xi − X∗
+i ln Xi),
+for some constants ci ≥ 0 to be determined [2, 5, 6, 20, 27, 29, 32, 33, 45, 49, 52, 53,
+59, 63, 65]. These functions are adapted from a first integral of the notorious Lotka-
+Volterra prey-predator system, and were popularized by Bean-San Goh in a series of
+paper [12, 13, 14].
+3. Quadratic. The Lyapunov function for the EE when R0 > 1 is of the common form
+L =
+�
+i
+ci(Xi − X∗
+i )2,
+for some constants ci ≥ 0 to be determined, or the composite form
+L =
+��
+i
+Xi − X∗
+i
+�2
+.
+Some examples can be found in [40, 41, 60, 65, 66].
+4. Integral Lyapunov. Lyapunov functions given as integrals over the dynamics of the
+model.
+The integration interval often start at some EE value X∗
+i and ends at the
+same Xi; this construction is very convenient if uniqueness of the EE is guaranteed,
+but the exact values of the EE are hard (or impossible) to determine analytically.
+Integral Lyapunov functions are particularly useful when the model includes multiple
+stages of infection, and consequently the infectious period changes from an exponential
+distribution to a gamma distribution [8, 11, 18, 38, 58, 61]. Integral Lyapunov functions,
+albeit in different forms, are widely used in models which incorporate explicit delay,
+such as systems of Delay Differential Equations (from here onwards, DDEs), and age-
+structured models. However, these fall beyond the scope of this paper, and we will
+briefly comment on them in Section 3.
+5. Hybrid.
+A linear combination of the above, which often includes the Goh-Lotka-
+Volterra in at least a few of the compartments of the system [15, 27, 37, 47, 50, 53, 51,
+63].
+3
+
+For some high-dimensional models, proving convergence to the EE might require addi-
+tional tools, such as the geometric approach used in [53, 64].
+Lastly, we must notice that not all compartmental models only exhibit convergence to
+equilibrium. Some systems of autonomous ODEs may present stable or unstable limit cycles
+[9, 54, 68], homoclinic orbits [54] or even chaos [57]. In such cases, clearly, no global Lyapunov
+function may exist.
+In the remainder of this survey, we will present various models and the corresponding
+Lyapunov functions, covering all the cases listed above.
+2
+Epidemic models
+In this section, we present various compartmental epidemic models with the corresponding
+Lyapunov function(s). We present the models from the smallest to the largest, in terms
+of number of compartments. We refer to [1, 28] for a basic introduction on compartmental
+epidemic models, and to [55] for a detailed exemplification of Lyapunov theory in this setting.
+We provide a schematic representation of the flows in most of the systems we present.
+Flow diagrams can be useful to provide a visual, intuitive interpretation of the parameters
+involved in each system. Arrows between compartments indicate a change in the current
+state of individuals with respect to the ongoing epidemics, whereas arrows inward/outward
+the union of the compartments represent birth rate and death rate in the population. Often,
+these last two rates are considered to be equal, as this assumption allows the population to
+either remain constant or converge to a constant value, reducing the dimensionality of the
+system and (hopefully) its analytical complexity. However, some models include additional
+disease-induced mortality, to increase realism when modelling severe infectious diseases. We
+uniform the notation throughout the various models we present in this survey as much as
+possible, and provide a brief description of each parameter the first time it is encountered.
+We remark that each variable is assumed to be non-negative, since it represents a fraction of
+the population, but the biologically relevant region varies depending on the specific model
+we are describing.
+Moreover, we illustrate the corresponding Lyapunov functions for 2D models, showcasing
+a selection of their power levels. The same procedure can be easily adapted to 3D models,
+but the corresponding visualizations can be hard to interpret in a static image.
+2.1
+SIS
+The SIS model is characterized by the total absence of immunity after infection, i.e. the
+recovery from infection is followed by an instantaneous return to the susceptible class. The
+ODEs system which describes this situation is
+dS
+dt = γI − βSI
+N ,
+dI
+dt = βSI
+N − γI,
+(1)
+S
+I
+β SI
+N
+γI
+where β is the transmission rate and γ is the recovery rate.
+4
+
+Notice that the population N = S + I is constant, thus we can normalize it to N = 1.
+Moreover, since S + I = 1, we can reduce the system to one ODE which involves only
+infectious individuals
+dI
+dt = (β(1 − I) − γ)I.
+System (1) always admits the DFE, i.e. E0 = (1, 0), and the EE, i.e. E∗ =
+�γ
+β , β − γ
+β
+�
+,
+which exists if and only if β > γ (or equivalently if R0 = β/γ > 1). Notice that, if R0 < 1,
+then I is always decreasing in the biologically relevant interval [0, 1].
+A variation of model (1) can be obtained by adding demography to the system. This is
+the example of [65], in which the authors consider a birth/immigration rate different from
+the natural death rate; moreover, they include an additional disease-induced death rate from
+infectious class. Thus, the population is not constant and the system of ODEs which describe
+the model is
+dS
+dt = Λ + γI − βSI
+N − µS,
+dI
+dt = βSI
+N − (δ + γ + µ)I,
+(2)
+S
+I
+β SI
+N
+γI
+Λ
+µS
+(δ + µ)I
+where Λ represents the birth/immigration rate, µ the natural death rate and δ the disease-
+induced mortality rate. System (2) always admits the DFE, namely E0 = (S0, 0) :=
+�Λ
+µ, 0
+�
+,
+and the EE, namely E∗ = (S∗, I∗), where I∗ > 0 if and only if R0 =
+Λβ
+µ(µ + δ + γ) > 1. In
+[65], a Lyapunov function for the DFE is defined as
+V (S, I) := 1
+2 (S − S0 + I)2 + 2µ + δ
+β
+I,
+(3)
+whereas the Lyapunov function for the EE is built using a combination of the quadratic and
+logarithmic functions
+V (S, I) := 1
+2 (S − S∗ + I − I∗)2 + 2µ + δ
+β
+�
+I − I∗ − I∗ ln
+� I
+I∗
+��
+.
+(4)
+The authors also construct two more examples of Lyapunov functions for the EE, namely
+V (S, I) := 1
+2(S − S∗)2 + µ + δ
+β
+�
+I − I∗ − I∗ ln
+� I
+I∗
+��
+,
+(5)
+and
+V (S, I) :=1
+2 (S − S∗ + I − I∗)2 + S∗(δ + 2µ)
+2γ
+�
+S − S∗ − S∗ ln
+� S
+S∗
+��
++ S∗(δ + 2µ)
+γ
+�
+I − I∗ − I∗ ln
+� I
+I∗
+��
+.
+(6)
+5
+
+Power levels of the functions (3), (4), (5) and (6) are visualized if Figure 1. By definition
+of a Lyapunov functions, orbits of the corresponding system (2) evolve on decreasing power
+levels, and they tend to the corresponding equilibrium as t → +∞.
+(a)
+(b)
+(c)
+(d)
+Figure 1: Power levels of Lyapunov functions (3) (a), (4) (b), (5) (c), and (6) (d). Values of
+the parameters are Λ = 0.8, µ = 1, δ = 1, γ = 1 in all the figures, β = 1 in (a), so that R0 =
+4/15 < 1, and β = 4 in (b), (c) and (d), so that R0 = 16/15 > 1. We represent V (S, I) = k,
+with k ∈ {0.1, 0.25, 0.5, 1, 1.5, 2, 2.5} in (a), k ∈ {0.001, 0.01, 0.025, 0.05, 0.1, 0.2} in (b) and
+(c), and k ∈ {0.01, 0.025, 0.05, 0.1, 0.2, 0.5} in (d). Black dots represent the globally stable
+equilibrium the system converges to, and correspond to V (S, I) = 0.
+In [66] the author found a simpler Lyapunov function for the DFE when R0 < 1, i.e.
+V (I) = 1
+2I2.
+(7)
+However, this last Lyapunov function (7) only ensures that I → 0 as t → +∞. To complete
+6
+
+the proof of the converge of the system to the DFE, one needs in addiction to invoke LaSalle’s
+theorem [35] (see also [31, Thm. 3.4]), as is indeed done in [66].
+Considering the importance of this theorem, especially when combined with the use of
+Lyapunov functions, we include its statement here.
+Theorem 2.1. (LaSalle’s invariance principle) Let X′ = f(X) be a system of n ODEs
+defined on a positively invariant set Ω ⊂ Rn.
+Assume the existence of a function V ∈
+C1(Ω, R) such that V ′(X) ≤ 0 for all X ∈ Ω. Let MV be the set of stationary points for V ,
+i.e. V ′(X) = 0 for all X ∈ MV , and let N be the largest invariant set of MV . Then, every
+solution which starts in Ω approaches N as t → +∞.
+In particular, this theorem implies that, if we can prove the approach of the disease to
+the manifold describing absence of infection and the uniqueness of the DFE, then the DFE
+is GAS.
+2.2
+SIR/SIRS
+The SIR model is characterized by the total immunity after the infections, i.e. recovered
+individuals can not become susceptible again. A classical example for this scenario is measles.
+The ODEs system which describes this situation is
+dS
+dt = −βSI
+N ,
+dI
+dt = βSI
+N − γI,
+dR
+dt = γI,
+(8)
+S
+I
+R
+β SI
+N
+γI
+where β is the transmission rate and γ is the recovery rate.
+If we assume that recovered individuals eventually lose their immunity, we obtain the
+SIRS model. Denoting by α the immunity loss rate, we obtain the following ODEs system
+dS
+dt = −βSI
+N + αR,
+dI
+dt = βSI
+N − γI,
+dR
+dt = γI − αR.
+(9)
+S
+I
+R
+β SI
+N
+γI
+αR
+It is clear that, if α = 0, system (9) coincides with system (8).
+These models admit only the DFE; in order to have an EE, we need to add the demog-
+raphy to model (8) or (9).
+In [65], the authors consider the following ODEs system
+7
+
+dS
+dt = Λ − βSI
+N − µS + αR,
+dI
+dt = βSI
+N − (γ + δ + µ)I,
+dR
+dt = γI − (α + µ)R.
+(10)
+S
+I
+R
+β SI
+N
+γI
+αR
+Λ
+µS
+µR
+(δ + µ)I
+System (10) admits the DFE, E0 = (S0, 0, 0), and the EE, E∗ = (S∗, I∗, R∗), which exists if
+and only if R0 =
+βΛ
+µ(µ + γ + δ) > 1. In [65], the Lyapunov function for the DFE is defined
+as follows
+V (S, I, R) := 1
+2 (S − S0 + I + R)2 + 2µ + δ
+β
+I + 2µ + δ
+2γ
+R2,
+whereas the Lyapunov function for the EE is the combination of the composite quadratic,
+common quadratic and logarithmic functions as follows
+V (S, I, R) :=1
+2 (S − S∗ + I − I∗ + R − R∗)2
++ 2µ + δ
+β
+�
+I − I∗ − I∗ ln
+� I
+I∗
+��
++ 2µ + δ
+2γ
+(R − R∗)2.
+The authors also present other Lyapunov functions for SIR/SIRS models; in particular,
+they also cite [3, 46] in which some variations of system (10) are showed. Other Lyapunov
+functions for SIR/SIRS epidemic models are in [55], in which the authors use a graph-
+theoretic approach.
+In [66], the author proved that the quadratic Lyapunov function (7) of the SIS model
+applies to the SIR and the SIRS, as well.
+2.3
+SEIR/SEIS/SEIRS
+In [32], the authors study both SEIR and SEIS models. Many real world examples present
+a phase of exposition to the disease, between susceptibility and infectiousness. The models
+presented thus far, albeit simpler to study, are unable to replicate this mechanism.
+The authors first analyze a SEIR model with demography and constant population, in
+which the disease is transmitted both horizontally and vertically. Individuals infected verti-
+cally pass first in the exposed compartment. The ODEs system which describe the model is
+dS
+dt =µ − βSI − pµI − qµE − µS,
+dE
+dt =βSI + pµI − θE − µE + qµE,
+dI
+dt =θE − (δ + µ)I,
+(11)
+S
+E
+I
+βSI
+θE
+µ(1 − pI − qE)
+µS
+µ(pI + qE)
+µE
+(δ + µ)I
+and R = 1 − S − E − I. The vertical transmission of the disease is represented by the
+probabilities p and q of being born directly in the Exposed compartment, rather than in the
+Susceptible one, and is represented by the inward arrow in compartment E.
+8
+
+The authors first provide an equivalent system, performing the substitution (S, E, I) −→
+(P, E, I), where P := S + pµ
+β . They then proceed to prove the GAS of the EE, using the
+following Lyapunov function
+V (P, E, I) :=(P − P ∗ ln P) +
+θ + µ
+θ + µ − qµ(E − E∗ ln E)
++
+θ + µ
+θ + µ − qµ(I − I∗ ln I).
+Later, the authors analyze a situation in which the recovery does not provide immunity,
+namely the SEIS model. They also assume that a fraction r of offspring of the infective
+hosts is born directly into the infective compartment. In this case, the ODEs system changes
+accordingly describe the model is
+dS
+dt =µ − βSI + (δ − pµ − rµ)I − qµE − µS,
+dE
+dt =βSI + pµI − (θ + µ − qµ)E,
+dI
+dt =θE − (δ + µ − µr)I,
+(12)
+and S + E + I = 1. Notice that, due to the population remaining constant in system (12),
+one could in principle reduce its dimensionality and consider it as a planar system.
+The authors prove the GAS of the EE using the following Lyapunov function
+V (S, E, I) :=(S − S∗ ln S) + µ1 − S∗
+βI∗S∗ (E − E∗ ln E)
++ µ1 − S∗
+θE∗
+�
+1 + pρ0
+µ
+β
+�
+(I − I∗ ln I).
+A natural extension to these models is the SEIRS [22, 64], in which one can combine the
+existence of an immune compartment and the loss of immunity.
+It is described by the
+following system of ODEs
+dS
+dt = − βg(I)S + µ − µS + αR,
+dE
+dt =βg(I)S − (θ + µ)E,
+dI
+dt =θE − (γ + µ)I,
+dR
+dt =γI − (α + µ)R,
+(13)
+S
+E
+I
+R
+βg(I)S
+γI
+αR
+θE
+µE
+µ
+µS
+µR
+µI
+where g ∈ C3(0, 1], g(0) = 0 (meaning, in absence of infectious individuals, the disease does
+not spread) and g(I) > 0 for I > 0. The classical choice is g(I) = I, as in systems (11) and
+(12). Assuming moreover
+lim
+I→0+
+g(I)
+I
+= c ∈ [0, +∞),
+9
+
+the authors of [22] derive R0 =
+cβθ
+(θ + µ)(γ + µ). They then prove GAS of the DFE of system
+(13) through the use of the following linear Lyapunov function
+V (E, I) = E + θ + µ
+θ
+I,
+whereas the GAS of the EE is proved with a more complex geometrical method in [64].
+2.4
+SAIR/SAIRS
+One of the main challenges of the Covid-19 pandemic was the presence of asymptomatic
+individuals spreading the disease. Such individuals must clearly be somehow distinguished
+from symptomatic infectious individuals, as they are likely to behave like a susceptible
+individual. Even though their viral load, and hence infectiousness, might be smaller, they
+are more likely to get in close contact with susceptible individuals.
+In [53], the authors consider a SAIRS model. The main difference between this kind
+of models and the SEIR is that both asymptomatic and symptomatic hosts may infect
+susceptible individuals.
+The immunity is not permanent, i.e.
+recovered individuals will
+become susceptible again after a certain period of time. Moreover, vaccination are included.
+The ODEs system which describe this model is
+dS
+dt = µ −
+�
+βAA + βII
+�
+S − (µ + ν)S + γR,
+dA
+dt =
+�
+βAA + βII
+�
+S − (α + δA + µ)A,
+dI
+dt = αA − (δI + µ)I,
+dR
+dt = δAA + δII + νS − (γ + µ)R,
+S
+A
+R
+I
+µ
+µS
+(βAA + βII)S
+δII
+γR
+δAA
+µA
+αA
+νS
+µI
+µR
+The global stability analysis of the EE has been performed for two variations of the original
+model, described in the following.
+The first model analyzed is the SAIR model, i.e. the case in which recovery from the
+disease grants permanent immunity. In this case, the corresponding Lyapunov function is
+the combination of the Lokta-Volterra Lyapunov functions for S, A and I
+V (S, A, I) :=c1S∗
+� S
+S∗ − 1 − ln
+� S
+S∗
+��
++ c2A∗
+� A
+A∗ − 1 − ln
+� A
+A∗
+��
++ I∗
+� I
+I∗ − 1 − ln
+� I
+I∗
+��
+,
+where c1, c2 > 0.
+The second model is the SAIRS model, with homogeneous disease transmission and
+recovery among A and I, i.e. βA = βI and δA = δI. In this case, it is possible to sum
+10
+
+equations for A and I, defining M := A + I, reducing the dimensionality of the system.
+Thus, the Lyapunov function can be written as the combination of the square function and
+the Lokta-Volterra as follows
+V (S, M) := 1
+2(S − S∗)2 + w
+�
+M − M∗ − M∗ ln
+� M
+M∗
+��
+,
+where w > 0.
+The global stability in the most general case is proved similarly to [64].
+2.5
+More exotic compartmental models
+The aforementioned models are some of the most commonly used in literature. In order
+to capture additional disease-specific nuances, these model can be modified or extended by
+adding new compartments.
+Some diseases, for example, present different stages of infection. In this case, an infected
+individual can progress between two or more stages before recovering. In [18], the authors
+perform the global stability analysis via an integral Lyapunov function of a general class
+of multistage models.
+In their model, infectious individual can move both forward and
+backward on the chain of stages, in order to incorporate both a natural disease progression
+and the amelioration due to the effects of treatments.
+The system of ODEs which describes the model is
+dS
+dt = θ(S) − f(N)
+n
+�
+j=1
+gj(S, Ij),
+dI1
+dt = f(N)
+n
+�
+j=1
+gj(S, Ij) +
+n
+�
+j=1
+φ1,j(Ij) −
+n+1
+�
+j=1
+φj,1(I1) − ζ1(I1),
+dIi
+dt =
+n
+�
+j=1
+φi,j(Ij) −
+n+1
+�
+j=1
+φj,i(Ii) − ζi(Ii),
+i = 2, 3, . . ., n,
+where θ(S) is the growth function, f(N)
+�n
+j=1 gj(S, Ij) is the incidence term, ζi(Ii), 1 ≤ i ≤ n,
+denote the removal rates of the Ii compartment.
+Moreover, for any i, j = 1, . . . , n, the
+functions φi,j(Ij) represent the rate of the disease progression if i > j and the amelioration
+if i < j.
+The corresponding Lyapunov function for the DFE is linear in the disease compartments,
+i.e.
+V (I1, . . ., In) =
+n
+�
+i=1
+ciIi,
+where c1 = R0 and ci ≥ 0 for all i = 2, . . . , n. For the global stability of the EE the authors
+made some assumptions on the aforementioned functions. In particular, they consider the
+following integral Lyapunov function
+V (S, I1, . . . , In) = τ
+� S
+S∗
+Φ(ξ) − Φ(S∗)
+Φ(ξ)
+dξ +
+n
+�
+i=1
+τi
+� Ii
+I∗
+i
+ψi(ξ) − ψi(I∗
+i )
+ψi(ξ)
+dξ,
+11
+
+where τ, τi > 0, for all i = 1, . . ., n. For a more in-depth explanation on the functions Φ(·)
+and ψi(·) we refer to [18, Sect. 5].
+Diseases which present multiple virus strains, due to the existence of different serotypes
+of the virus or due to a mutation of the original disease, may need to be modelled differently.
+Dengue, tuberculosis and various sexually transmitted diseases are caused by more than one
+strain of a pathogen. Influenza type A viruses mutate constantly: an infection with one of
+its strains gives permanent immunity against that specific strain. However, the so called
+“antigenic drift” produces new virus strains, thus the hosts only acquire partial immunity, or
+no immunity at all. Modelling these types of diseases requires the inclusion of cross-protective
+effects, in which the immunity acquired towards one strain offers partial protection towards
+another strain based on their antigenic similarity. In [6], the authors consider an n strain
+model, both without immunity and with immunity for all the strains. Moreover, they analyze
+an MSIR model, in which the M compartment represents the proportion of newborns who
+possess temporary passive immunity due to protection from maternal antibodies. For all the
+three model, the authors use a linear Lyapunov function to prove the global stability of the
+DFE and a logarithmic Lyapunov function to prove the global stability of the EE.
+Other compartmental models include e.g. control strategies. For new ongoing epidemics,
+the most immediate strategy is including quarantine and isolation of infectious individuals.
+For well-known epidemics for which a vaccination is available, it is useful to incorporate a
+vaccinated individuals compartment V to keep track of the two possible immunities, disease
+and vaccine induced, respectively. Usually, vaccination does not confer permanent immu-
+nity, and after a certain disease-dependent period individuals become susceptible again. An
+example is [50], in which the authors analyze a SIRV epidemic model with non-linear inci-
+dence rate. The global stability of the DFE is proved using as linear Lyapunov function the
+infectious compatment I and the global stability of the EE, instead, using a combination of
+a quadratic function in S and a logarithmic function in the compartments I and V .
+3
+Conclusion
+In this survey, we presented the most widely used Lyapunov functions in the field of epi-
+demic compartmental models. We focused on systems expressed as autonomous systems of
+ODEs. These models allow for various interesting generalizations, of which we provide a
+non-comprehensive list below.
+One extension of the classic compartmental epidemic models is the so-called multi-group
+approach, see e.g. [34, 58]. These models describe n communities, interacting with each other,
+and whose internal evolution follows a standard compartmental model. A first example of
+such a model is presented in [10], in which the authors consider a n groups SIS model. In
+order to prove the GAS of the EE, they use a results on Metzler matrices. In [55], the authors
+consider a heterogeneous SIS disease model, for which they provide Lyapunov functions both
+for the DFE and for the EE. For the latter, they use a complex graph-teoretic method, for
+the details of which we refer to the original paper. Global stability of EE via Lyapunov
+function for multi-group generalization can be found also for the SIR [19], SIRS [48], SEIR
+[17] and SAIR/SAIRS model [52]. Notice that, due to the complexity of the models, some
+of them require additional technical assumptions to prove the global stability of the endemic
+equilibrium.
+12
+
+Other classes of models include interactions between human and vector population, i.e.
+animals which transmit the disease to humans, or with the pathogens, such as viruses or
+bacteria. In both cases, authors often include a compartmental structure for the non-human
+population. Some examples of vector-host models are shown in [59, 62, 70]. Another example
+can be found in [40], in which a SIR-B compartmental model is considered. Here the “B”
+denotes the concentration of the pathogen in the environment.
+All the models discussed thus far are described by only autonomous systems of ODEs.
+However, in order to increase realism, it is possible to use non-autonomous systems to de-
+scribe the spread of an infectious disease. This is the case of systems in which some param-
+eters change in time [42, 56], to describe seasonal changes, or in which the state variables
+depend on the previous state, i.e. the model includes a time delay [4, 67]. In these cases,
+it is still possible to find Lyapunov functions to prove the global stability of the equilibria
+using other techniques, described for example in [35].
+Another popular option is to explicitly include delay in the system, such as in [4, 23,
+25, 26, 43, 63, 69].
+In the latter the authors perform the global stability analysis of a
+SEIQR model, in which Q denotes the quarantined individuals. They explicitly include a
+latent period for the infection, transforming two of the ODEs in DDEs. The corresponding
+Lyapunov function includes the integration over an interval whose size is precisely the latent
+period.
+Lastly, a widely adopted strategy is to explicitly include the “time since infection” [7,
+24, 44, 71, 72] in age-structured models. This allows to explicitly take into account time
+heterogeneity in the spread of an infectious disease in a population.
+These last cases we mentioned are outside of the scope of this project, and we leave them
+as inspiration for future works.
+Acknowledgments.
+The authors are grateful to the organizers of the conference 100 Years
+Unione Matematica Italiana - 800 Years Università di Padova, which made their scientific
+cooperation possible. Moreover, they acknowledge Politecnico di Milano, Polish Academy of
+Sciences, Inria and University of Trento for supporting their research.
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+

2tFAT4oBgHgl3EQfDhzt/content/tmp_files/2301.08417v1.pdf.txt

@@ -0,0 +1,727 @@
+Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback
+Suppression of laser beam’s polarization and intensity fluctuation via a
+Mach-Zehnder interferometer with proper feedback
+Xiaokai Hou,1 Shuo Liu,1, a) Xin Wang,1 Feifei Lu,1 Jun He,1, 2 and Junmin Wang1, 2, b)
+1)State Key Laboratory of Quantum Optics and Quantum Optics Devices, and Institute of Opto-Electronics, Shanxi University,
+Tai Yuan 030006, Shanxi Province, China
+2)Collaborative Innovation Center of Extreme Optics, Shanxi University, Tai Yuan 030006, Shanxi Province,
+China
+(Dated: 23 January 2023)
+Long ground-Rydberg coherence lifetime is interesting for implementing high-fidelity quantum logic gates, many-body
+physics, and other quantum information protocols. But, the potential well formed by a conventional far-off-resonance
+red-detuned optical-dipole trap that is attractive for ground-state cold atoms is usually repulsive for Rydberg atoms,
+which will result in the rapid loss of atoms and low repetition rate of the experimental sequence. Moreover, the coher-
+ence time will be sharply shortened due to the residual thermal motion of cold atoms. These issues can be addressed
+by an one-dimensional magic lattice trap and it can form a deeper potential trap than the traveling wave optical dipole
+trap when the output power is limited. And these common techniques for atomic confinement generally have certain
+requirement on the polarization and intensity stability of the laser. Here, we demonstrated a method to suppress both
+the polarization drift and power fluctuation only based on the phase management of the Mach-Zehnder interferometer
+for one-dimensional magic lattice trap. With the combination of three wave plates and the interferometer, we used the
+instrument to collect data in the time domain, analyzed the fluctuation of laser intensity, and calculated the noise power
+spectral density. We found that the total intensity fluctuation composed of laser power fluctuation and polarization drift
+was significantly suppressed, and the noise power spectral density after closed-loop locking with typical bandwidth
+1-3000 Hz was significantly lower than that under the free running of the laser system. Typically, at 1000 Hz, the
+noise power spectral density after locking was about 10 dB lower than that when A Master Oscillator Power Amplifier
+(MOPA) system free running. The intensity-polarization control technique provides potential applications for atomic
+confinement protocols that demand for fixed polarization and intensity
+I.
+INTRODUCTION
+For various atomic manipulation experiments, such as sin-
+gle photon source1−5, quantum dynamics based on Rydberg
+states 6−10 and electric field detection based on atoms 11−13,
+strong confinement optical dipole trap (ODT) of atoms is usu-
+ally employed. In these applications,high power laser with
+fixed polarization and relatively stabled intensity normally is
+used to confine atoms. Common experimental setup for the
+laser power stabilization were based on the active feedback
+loop which used acousto-optic modulator (AOM) 14−17 or
+electro-optic modulator (EOM) 18 as the actuator. In 2020,
+AOM and EOM were combined to broaden the bandwidth of
+laser intensity noise stabilization to 1MHz by Ni et. al.19. At
+present, the feedback loop based on AOM has some disadvan-
+tages. For example, bragg diffraction of AOM will seriously
+affect the spot quality of first-order diffraction light, and the
+power utilization of the system will be limited by the diffrac-
+tion efficiency of AOM. The common electro-optic intensity
+modulator (EOIM) with input and output tailed fiber is effi-
+cient, but not suitable for high-power applications. Moreover,
+the schemes mentioned above can observably suppress the
+power fluctuation of laser beam, but the reduction for the drift
+a)Present Address: Key Laboratory of Laser & Infrared System of Ministry
+of Education, Shandong University, Qing Dao 266000, Shandong Province,
+China.
+b)corresponding author. E-mail: [email protected] ORCID : 0000-0001-
+8055-000X
+of laser’s polarization is still not be effectively achieved . Here
+we demonstrate an experimental scheme based on the Mach-
+Zehnder interferometer (MZI) for actively suppress both the
+fluctuation of power and polarization of laser beam. By prop-
+erly manipulated phase difference between two paths, the out-
+put fraction of MZI account for the majority laser power while
+its intensity fluctuation in the time domain has been reduced
+dozens of times compared with the free-running case, and the
+noise power spectral density (NPSD) has been decreased in
+the range of 1-3000 Hz in the frequency domain. Such a sta-
+ble system can certainly meet the needs of various applica-
+tions, such as experiments where the lifetime of cold atoms is
+highly desirable.
+II.
+THEORETICAL BACKGROUND
+1.
+Magic optical dipole trap for cesium 6S1/2 ground state
+and 84P3/2 Rydberg state
+Recently, a new experimental scheme which used interfer-
+ometer as the actuator of the feedback loop has been proposed
+20. Considering the light intensity requirement of the ODT,
+the MZI can satisfy the power requirement of ODT without af-
+fecting spot quality of output light, therefore the experimental
+setups of constructing the blue-detuning optical trap reported
+by Yelin et. al 21 and Isenhower et. al 22 both concentrate on
+the MZI. The intensity of the output laser mainly depends on
+the phase difference between two arms of the MZI, therefore it
+can be used as a power stabilizer in some experiments 23. Due
+arXiv:2301.08417v1  [physics.optics]  20 Jan 2023
+
+Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback
+2
+to the particularity of the output fraction of MZI, the com-
+bination of interferometer and polarizer can realize the fixed
+polarization, high proportion output and high intensity stabil-
+ity. It is obviously useful for the experiment of optical trap.
+The potential of ODT U can be expressed as:
+U = − α
+2ε0c
+2P
+πω2
+0
+(1)
+Where α is the induced polarizability of the target state, ε0
+is the permittivity of vacuum, c is the speed of light, P is the
+intensity of laser, ω0 is the radius of the spot at the focal point
+after the laser is focused by a lens . As shown in the Eq. (1),
+if the power of the 1879 nm laser is fluctuant, the resulting
+trap depth will be changed. Thus, the lifetime of the trapped
+atom will be severely affected by the presence of the heating
+mechanism5,17,24.
+FIG. 1. Diagram of the light shift induced by the ODT and MODT.
+The intensity of laser which is intensly focused is still Gaussian,
+and the closer to the center of beam, the stronger the intensity of
+laser. The resulting trap depth or light shift is spatially dependent
+(a) The ODT is attractive for ground states, but usually repulsive
+for highly-excited Rydberg states because almost all strong dipole
+transitions connected Rydberg state and the lower states have longer
+wavelength than that of ODT laser. (b) The direct single-photon ex-
+citation scheme from cesium |g⟩=|6S1/2⟩ to |r⟩=|84P3/2⟩ coupled by
+a 319 nm ultraviolet laser. A 1879.43 nm laser is also tuned to the
+blue side of the |r⟩ ⇐⇒ |a⟩=|7D5/2⟩ auxiliary transition to equalize
+the trapping potential depth of the |g⟩ and |r⟩ state, which is so called
+magic ODT (MODT).
+In most of the experiments of cold atoms involving confine-
+ment of ground-state atoms in an ODT and Rydberg excita-
+tion, cold atomic sample is prepared in an ODT to hold them
+in a fixed position in a significantly long time. The poten-
+tial formed by a conventional far off-resonance red-detuned
+ODT is attractive for the ground-state atoms, but usually re-
+pulsive for highly-excited Rydberg atoms, leading that Ryd-
+berg atoms normally cannot be confined in the conventional
+ODT (Fig. 1(a)). Therefore, in the follow-up experiments, we
+will face the following two problems: (1) if switching off the
+ODT during Rydberg excitation and coherent manipulation, it
+will result in atomic dephasing due to the thermal diffusion
+of the atoms and the extremely low repetition rate of the ex-
+perimental sequence; (2) if the ODT remains operation, it may
+cause a low Rydberg excitation efficiency of atoms as the tran-
+sition frequency is spatially position-dependent on the excita-
+tion laser. The solution is to find an ODT such that the ground-
+state atoms and the desired highly-excited Rydberg atoms can
+experience the same potential, that is, the potential generated
+by the ODT is a potential well for both the ground-state atoms
+and the desired highly-excited Rydberg atoms, and is attrac-
+tive to atoms in both states. So, the above-mentioned aspects
+(1) and (2) can be solved. In Fig.1(b), the direct single-photon
+excitation scheme from cesium |g⟩=|6S1/2⟩ to |r⟩=|84P3/2⟩
+coupled by a 319 nm ultraviolet laser. A 1879.43 nm laser
+is also tuned to the blue side of the |r⟩ ⇐⇒ |a⟩=|7D5/2⟩ aux-
+iliary transition to equalize the trapping potential depth of the
+|g⟩ and |r⟩ state. The specific calculation process is not de-
+scribed here. For details, please refer to the reference 25,26.
+2.
+Theoretical analysis of MZI
+It is obvious that the MODT is not enough to meet the
+need of extremely long coherence time in subsequent experi-
+ments. The cold atoms trapped in the MODT still have resid-
+ual thermal motion, which causes violent collisions that heat
+the atoms and cause them to escape from the trap. We will fur-
+ther construct one-dimensional magic lattice trap (1D-MLT),
+and combine the advantages of lattice and magic conditions,
+so as to prolong the coherence time of the ground-Rydberg
+state of cold atoms. Of course, the 1D-MLT also needs to
+suppress its power fluctuation. Because the power of the laser
+used in the 1D-MLT fluctuates in the time domain, will di-
+rectly shorten the coherence lifetime of the cold atom. There-
+fore, we use the MZI to suppress the power ���uctuation.
+As shown in Fig. 2 (a), Iout1 and Iout2 are the intensity of
+two output paths of the interferometer respectively; R1, T1, R2,
+T2 are the reflectivity and transmittance of input and output
+beam splitters plate respectively. The two output channels of
+the interferometer can be expressed as Eq. (2) and (3):
+Iout1 = R2
+1R2
+2 +T 2
+1 T 2
+2 +2R1R2T1T2cos(2∆L
+λ
++π)
+(2)
+Iout2 = R2
+1R2
+2 +T 2
+1 T 2
+2 +2R1R2T1T2cos(2∆L
+λ )
+(3)
+Therefore, the laser intensity output of the interferometer can
+be controlled by adjusting the driving voltage of PZT due to
+the correlation between the output transmittance I and optical
+path difference ∆L. In Fig. 2 (b), the interference fringes
+generated by splitters with different splitter ratio is simulated
+and analyzed by Mathematica. The splitter ratio shown by the
+first line of Fig. 2(b) is 90/10, the second is 70/30, the third is
+60/40, and the last is 50/50.
+III.
+EXPERIMENTAL SETUP
+The laser intensity stabilization setup is shown in Fig. 3. A
+MOPA system consists of a 1879-nm butterfly packaged laser
+diode and a Thulium Doped Fiber Amplifier (TmDFA) which
+has maximum output ∼ 3 W. With a free space polarization
+
+I(r
+84P
+3/2
+1879.43nm
+319nm
+ODT
+1879.43nm
+g
+(a)
+6S1/2
+(b)Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback
+3
+FIG. 2. Diagram of MZI and interference fringes of two channels of
+the MZI are simulated and analyzed theoretically. (a) MZI consists
+of two beam splitter plates (BS1 and BS2) and two high-reflectivity
+mirrors (M1 and M2). Iin is the intensity of the incident light field,
+Iout1 and Iout2 are the intensity of the outgoing light field at BS2. (b)
+Normalized signal as a function of difference of optical path ∆L for
+different splitter ratio.This ratio is both R1/T1 and R2/T2, because
+the BS1 and BS2 that are used in the MZI are same. The solid red
+and black lines represent the interference fringes of the two output
+channels of MZI, respectively.
+controller based on three waveplates ( λ/4, λ/2 and λ/4 ),
+polarization fluctuation of 1879 nm beam is suppressed ini-
+tially. The laser is injected into a MZI which is constructed by
+a 50/50 beam splitter plate (BS1) that divides the incident light
+into two beams with equal intensity and the different phase, a
+high-reflectivity mirror (M1) that reflects one beam, a mirror
+(M2) attached to a PZT that emits the other, and a beam split-
+ter plate (BS2) that the two beams are finally combined. The
+interferometer has two output channels and each channel can
+be used for dynamic feedback to make the system more sta-
+ble, and the output of this channel can then be used for sub-
+sequent experiments. The photodetector (PD1) is mounted
+behind a glass slice (GS1) of 1879 nm for sampling a little
+fraction of light for in-loop feedback. The DC voltage signal
+output by the PD1 is injected into Proportional Integral Dif-
+ferential (PID) amplifier after passing through a low-pass filter
+(LPF). The input signal of PID controller is subtracted from
+the PID Set Point, which is an artificially set reference DC
+voltage. The output signal of PID, that is the real-time differ-
+ence between the detector signal and the reference DC volt-
+age is added with the scanning signal (triangular wave) and
+amplified by the high voltage (HV) amplifier as the driving
+voltage of the PZT. The output power of interferometer can
+therefore be controlled by manipulating the driving voltage of
+PZT, and we expect that both power and polarization fluctua-
+tion for 1879 nm laser are suppressed. And another photode-
+tector (PD2) is mounted in order to independently monitor the
+intensity stability of the output linear polarization laser. The
+output signal of PD2 is then injected into the Data Acquisition
+System (Keithley, DAQ-6510) in order to analyze and moni-
+tor the intensity fluctuation of the laser in the time domain and
+calculate the NPSD based on the measured optical power fluc-
+tuation data. Undoubtedly, the little fraction of the far-infrared
+laser is reflected by glass slice (GS2) and received by the PD2
+and the majority of laser is transmitted and focused in a ce-
+sium magneto-optical trap (Cs-MOT) for the construction of
+the ODT.
+FIG. 3. Experimental setup for intensity stabilization system. The
+dynamic stability of laser intensity of 1879 nm MOPA system is re-
+alized by MZI, and the fluctuation of laser intensity is monitored
+and analyzed in time domain and frequency domain.λ/2: half-wave
+plate; λ/4: quarter-wave plate; PBS: polarization beam splitting
+cube; BS: beam splitting plate; GS: glass slice; M1/M2: high-
+reflectivity mirror; PD: photodetector; LPF: low-pass filter; PID:
+Proportional Integral Differential amplifier; HVA: high voltage am-
+plifier.
+IV.
+EXPERIMENTAL RESULTS AND DISCUSSION
+Fig. 4 shows, the interference fringes obtained by scanning
+triangular waves with 50/50 beam splitter ratio in the experi-
+ment, in which the interference contrast is 95%. In theoretical
+simulation, an interference fringe with an interference con-
+trast of 99.9% can be obtained by using a 50/50 beam splitter
+plate, but the best interference contrast is not achieved in ex-
+periment, probably due to the following two reasons: first, the
+spatial mode of the two lasers is not exactly same; second, the
+polarization of the two lasers may be slightly different.
+Considering the requirement of constructing dipole trap
+with this laser source, the polarization of 1879 nm laser should
+be fixed, so PBS is usually inserted in the light path to fixed
+the polarization of light. Even though the scheme is effective,
+an inevitable defect exists in this scheme is that the polariza-
+tion fluctuation of light will couple with the intensity fluctu-
+ation through this polarization element. As the measurement
+of which the intensity for 1879 nm laser after a PBS, although
+the power fluctuation of 1879 nm TmDFA itself is not obvi-
+ous, the intensity fluctuation behind the PBS becomes obvious
+and the results is shown in Fig. 5(a). We monitor the laser in-
+tensity for about 30 minutes in the time domain, with a large
+fluctuation of about ±14.2%. The huge intensity fluctuation
+
+90/10
+70/30
+60/40
+50/50LSuppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback
+4
+FIG. 4. Interference fringe of the MZI. In the experiment, a 50/50
+beam splitter plate is used, and the PZT is driven by scanning tri-
+angular wave, so that the phase difference between the two arms is
+generated, then the interference fringes are generated.
+will significantly affect the power utilization of the stable sys-
+tem. To maximize the power utilization, three wave plates
+are used to suppressed the power fluctuation initially. After
+proper adjustment, measurement result of laser intensity fluc-
+tuation after PBS is shown in Fig. 5 (b). Fluctuation of laser
+polarization has been reduced significantly. Then the initial-
+stabled laser has been injected in the combine system of MZI
+and another PBS, here the transmittance of the interferometer
+is locked up to 90% in order to improve the power utilization.
+Then the intensity fluctuation probed by the out-of-loop detec-
+tor PD2 is shown in Fig. 6. As shown below, the intensity fluc-
+tuation of output linear polarized laser is reduced to ±0.3%,
+that is much better than the fluctuation of direct TmDFA-PBS
+output. At this stability, both fluctuation of laser power and
+polarization will no longer have a significant influence on the
+parameter of dipole trap.
+As shown in Table 1, for the 1879 nm 1D-MLT, if the laser
+is focused through a lens to ∼ 20 µm. and the incident laser
+power at the cold atom is about 1.5 W, so the maximum depth
+of the 1D-MLT is −1000 µK and the typical trap depth fluc-
+tuation is ±140 µK. When the laser power decreases after the
+initial suppression of the wave plate group or the closed-loop
+locking of the MZI, the corresponding typical trap depth is
+about −800 µK and −700 µK respectively. And the effec-
+tive temperature of the cold atoms which are transferred from
+MOT to 1D-MLT will be slightly higher, about 100 µK, but
+the decrease of trap depth caused by the suppression of power
+fluctuation will not affect the capture of the cold atoms. How-
+ever, the residual fluctuation of laser power still exists, which
+will lead to the typical trap depth fluctuation of ±45 µK and
+±2 µK respectively.
+The collected time-domain voltage signals are used to cal-
+culate the NPSD. As shown in Fig. 7, the horizontal range
+is determined by the sampling rate. In the experiment, we
+selected sampling rate of 10000 Hz according to the actual
+situation, so the horizontal axis in Fig. 7 ranges from 1 to
+5000Hz. In addition, we believe that the feedback bandwidth
+of the system should be at the level of kilohertz due to the
+limitation of PZT in the MZI. Therefore, the sampling rate
+can fully meet the requirement of representing the feedback
+bandwidth of the system.
+The NPSD after closed-loop locking from 1-3000Hz is sig-
+nificantly lower than that under the free running of the MOPA
+system. It can be proved that the MZI plays an obvious role in
+the power stability of the system. In order to further broaden
+the feedback bandwidth and improve the inhibitory effect, we
+assume that the arm length of the MZI is L and the angular fre-
+quency of the laser is ω0, then the distance of the laser going
+through the MZI is L and the phase shift generated is27
+Φ0(t) = ω0t = ω0
+L
+c
+(4)
+Φ0 is a constant, and the magnitude is proportional to L .
+When the PZT is scanned, we introduce to characterize small
+changes in phase. For simplicity, we assume that a sine wave
+is used to scan the PZT, and the amplitude of the sine wave is
+h0 and the angular frequency is ωs, so the sine wave can be
+expressed as
+h(t) = h0cos(ωst)
+(5)
+So, the phase shift of the entire system can be written as
+Φ = Φ0(t)+δφ
+= ω0L
+c
++ ω0
+2
+� t
+t�� L
+c
+h0cos(ωst)dt
+= ω0L
+c
++ h0
+2
+ω0
+ωs
+�
+sin(ωs
+L
+c )−sin[ωs(t − L
+c )]
+�
+= ω0L
+c
++h0
+ω0
+ωs
+sin(ωs
+L
+2c)cos[ωs
+2 (t − L
+c )]
+(6)
+because L
+2c ≪ 1
+h0
+ω0
+ωs
+sin(ωs
+L
+2c) = h0ω0
+2
+L
+c
+(7)
+δφ ∼ h0ω0
+2
+L
+c
+(8)
+As shown in Eq. (8), if the arm length L of the MZI is
+increased, δφ of the system can be increased. Thus, the de-
+tection sensitivity of the system can be improved and the de-
+tection effect of the MZI for phase can be better. Increasing
+the arm length of the MZI will cause extra noise due to the
+insufficient stability of the system.
+However, such noise can be solved through the isolation
+platform and system temperature control. We can add F-P
+cavity on the two arms of the MZI. F-P cavity can fold up the
+optical path, greatly increase the distance of light in the MZI,
+and do not need to occupy a large area.
+
+0.35
+0.28
+Locked point
+Voltage (V)
+0.21
+0.14
+0.07
+0.00
+0.00
+0.02
+0.04
+0.06
+0.08
+Time (s)Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback
+5
+FIG. 5. (a) The power fluctuation of free running 1879 nm MOPA system . Through 30 mins of measurement, the power fluctuation is roughly
+±14.2%. The inset is zoomed in on the vertical axis to 2.00∼3.00 W, and shows the intensity fluctuation in 30 mins. (b) The power fluctuation
+after three wave plates. We thought that the polarization fluctuation is initially suppressed by these plates. Similarly, after 30 minutes of
+measurement, the intensity fluctuation is approximately ±5.7%. And, the vertical axis range of the inset becomes 1.75∼2.25 W, the range of
+horizontal axis is still 0∼30 mins.
+Category
+PODT (mW)
+∆P(mW)
+Gaussian radius after focused (µm) Udip(µK) ∆Udip(µK)
+MOPA free running
+1500
+±213.0 (±14.2%)
+20
+−1000
+±140
+With wave plate group
+1200
+±68.4 (±5.7%)
+20
+−800
+±45
+After MZI is locked
+1100
+±3.3 (±0.3%)
+20
+−700
+±2
+TABLE I. The typical maximum trap depth and fluctuation of 1879.43nm 1D-MLT for cesium atoms under different power fluctuations are
+calculated.
+FIG. 6. The intensity fluctuation of 1879 nm laser on the bright fringe
+of the MZI. By inter-of-loop locking, the phase difference between
+the two arms is dynamically compensated , and the power fluctuation
+is significantly suppressed , through 30 mins of measurement, the
+power fluctuation is roughly ±0.3%. The vertical axis of the inset
+has been enlarged with a range of 1.85∼1.88 W, and shows the 30-
+min measurement.
+FIG. 7. Intensity noise of 1879 nm laser as a function of analyze
+frequency. (a): The solid black line represents the NPSD when the
+1879nm laser system is running freely without passing through the
+wave plate group. (b): The solid blue line represents the NPSD of
+the 1879nm laser system after closed-loop locking by the MZI.
+
+3.00
+3.00
+2.50
+2.50 F
+Power fluctuation with wave plates group : ±5.7%
+Power fluctuaticn when MOPA free running : ±14.2%
+2.00
+2.00
+(W)
+M
+3.00
+2.25
+Power (
+Power (
+1.50
+1.50
+Power (w)
+()
+Power (
+2.50
+2.00
+1.00
+1.00
+0.50
+0.50
+2.00
+(a)
+1.75 ,
+(b)
+15
+20
+25
+30
+20
+25
+30
+T ime (min)
+Time (min)
+0.00
+0.00
+5
+10
+15
+20
+25
+30
+0
+5
+10
+15
+20
+25
+30
+0
+Time (min)
+Time (min)3.00
+2.50
+Power fluctuation after Mzl is locked : ±0.3%
+Powewr (W)
+2.00
+1.88
+1.50
+1.00
+1.86
+0.50
+1.85,
+10
+15
+20
+25
+30
+Time (min)
+0.00
+0
+5
+10
+15
+20
+25
+30
+Time (min)100
+3000Hz
+10
+NPSD (W/NHz)
+10
+(b)
+10-8
+MOPA free runming
+After MzI is locked
+10-9
+100
+101
+102
+103
+Frequency (Hz)Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback
+6
+V.
+CONCLUSIONS
+In summary, we have demonstrated the reduction of the
+power and polarization fluctuation for 1879 nm laser based
+on the cooperation of three wave plates and a MZI. The in-
+tensity fluctuation ∼ ±14.2% after the combination of MOPA
+system and PBS is reduced to ∼ ±0.3% with locked MZI.
+And after MZI is locked, the NPSD is lower than that under
+free running in the range of 1-3000 Hz. Typically, at 1000 Hz,
+the NPSD after MZI is locked is about 10 dB lower than that
+when MOPA free running. The system can not only withstand
+high power injecting laser, but also can stabilize both power
+fluctuation and polarization fluctuation without affecting the
+quality of light beam for the low-loss output light. The laser
+power utilizing efficiency can be further improved by improv-
+ing the transmittance of locked interferometer or improving
+the interference visibility.
+It is expected that Rydberg atoms can have long coherence
+lifetime in subsequent experiments involving Rydberg dressed
+ground state. On one hand, we can use the 1879-nm MOPA
+system to implement a 1D-MLT, which can both eliminate the
+position-dependent light shift to capture Rydberg-state atoms
+in optical tweezer like the ground-state atoms and attenuate
+collisions between cold atoms caused by residual thermal mo-
+tion to prolong the coherence time of the Rydberg atoms. On
+the other hand, we propose an upgraded interferometer, that
+is, adding a F-P cavity to each arm of the interferometer, and
+using the reflection of beam in the cavity, the arm length can
+be extended at least dozens of times, to improve the phase
+measurement sensitivity of the interferometer and improve the
+power stability.
+FUNDING
+This research was financially funded by the National Key R
+& D Program of China (2021YFA1402002), the National Nat-
+ural Science Foundation of China (11974226, and 61875111).
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+1
+Optimal Power Flow Based on
+Physical-Model-Integrated Neural Network with
+Worth-Learning Data Generation
+Zuntao Hu, Graduate Student Member, IEEE, and Hongcai Zhang, Member, IEEE
+Abstract—Fast and reliable solvers for optimal power flow
+(OPF) problems are attracting surging research interest. As
+surrogates of physical-model-based OPF solvers, neural network
+(NN) solvers can accelerate the solving process. However, they
+may be unreliable for “unseen” inputs when the training dataset
+is unrepresentative. Enhancing the representativeness of the
+training dataset for NN solvers is indispensable but is not well
+studied in the literature. To tackle this challenge, we propose an
+OPF solver based on a physical-model-integrated NN with worth-
+learning data generation. The designed NN is a combination
+of a conventional multi-layer perceptron (MLP) and an OPF-
+model module, which outputs not only the optimal decision
+variables of the OPF problem but also the constraints violation
+degree. Based on this NN, the worth-learning data generation
+method can identify feasible samples that are not well generalized
+by the NN. By iteratively applying this method and including
+the newly identified worth-learning samples in the training set,
+the representativeness of the training set can be significantly
+enhanced. Therefore, the solution reliability of the NN solver
+can be remarkably improved. Experimental results show that the
+proposed method leads to an over 50% reduction of constraint
+violations and optimality loss compared to conventional NN
+solvers.
+Index Terms—Optimal power flow, physical-model-integrated
+neural network, worth-learning data generation
+I. INTRODUCTION
+O
+PTIMAL power flow (OPF) is a fundamental but chal-
+lenging problem for power systems [1]. A typical OPF
+problem usually involves determining the optimal power dis-
+patch with an objective, e.g., minimizing total generation costs
+or power loss, while satisfying nonlinear power flow equations
+and other physical or engineering constraints [2]. Due to the
+nonlinear interrelation of nodal power injections and voltages,
+OPF is non-convex, NP-hard, and cannot be solved efficiently
+[3]. With the increasing integration of renewable generation
+and flexible demands, uncertainty and volatility have been
+rising on both the demand and supply sides of modern power
+systems [4], which requires OPF to be solved more frequently.
+Thus, fast and reliable OPF solvers have become indispensable
+to ensure effective operations of modern power systems and
+have attracted surging interest in academia.
+There is a dilemma between the solving efficiency and
+solution reliability of OPF. Conventionally, OPF is solved
+by iterative algorithms, such as interior point algorithms,
+based on explicit physical models [5]. However, these methods
+may converge to locally optimal solutions. Recently, some
+researchers have made great progress in designing conic
+relaxation models for OPF, which are convex and can be
+efficiently solved [6]–[8]. Nevertheless, the exactness of these
+relaxations may not hold in practical scenarios, and they may
+obtain infeasible solutions [9]. In addition, the scalability of
+the conic relaxation of alternating current optimal power flow
+(AC-OPF) may still be a challenge, particularly in online,
+combinatorial, and stochastic settings [10].
+To overcome the limitation of the aforementioned physical-
+model-based solvers, some researchers propose surrogate OPF
+solvers based on neural networks (NNs) [11]–[13]. These
+solvers use NNs to approximate the functional mapping from
+the operational parameters (e.g., profiles of renewable gen-
+eration and power demands) to the decision variables (e.g.,
+power dispatch) of OPF. Compared to iterative algorithms,
+they can introduce significant speedup because an NN is only
+composed of simple fundamental functions in sequence [12],
+[13]. However, one of the critical problems of NN solvers is
+that they may be unreliable if not properly trained, especially
+for “unseen” inputs in feasible regions due to NNs’ mystery
+generalization mechanism [14].
+The generalization of NNs is mainly influenced by their
+structures, loss functions, and training data. Most published
+papers propose to enhance the generalization of NN OPF
+solvers by adjusting the structures and loss functions. Various
+advanced NN structures rather than conventional fully con-
+nected networks are employed to imitate AC-OPF. For exam-
+ple, Owerko et al. [15] use graph NNs to approximate a given
+optimal solution. Su et al. [16] employ a deep belief network to
+fit the generator’s power in OPF. Zhang et al. [17] construct a
+convex NN solving DC-OPF to guarantee the generalization of
+NNs. Jeyaraj et al. [18] employ a Bayesian regularized deep
+NN to solve the OPF in DC microgrids. Some researchers
+design elaborate loss functions that penalize the constraints
+violation, combine Karush-Kuhn-Tucker conditions, or include
+derivatives of decision variables to operational parameters. For
+example, Pan et al. [11] introduce a penalty term related to the
+inequality constraints into the loss function. This approach can
+speed up the computation by up to two orders of magnitude
+compared to the Gurobi solver, but 18.3% of its solutions are
+infeasible. Ferdinando et al. [12] include a Lagrange item in
+the loss function of NNs. Their method’s prediction errors
+are as low as 0.2%, and its solving speed is faster than DC-
+OPF by at least two orders of magnitude. Manish et al. [10]
+include sensitivity information in the training of NN so that
+only using about 10% to 25% of training data can attain the
+same approximation accuracy as methods without sensitivity
+information. Nellikkath et al. [19] apply physics-informed
+arXiv:2301.03766v1  [cs.LG]  10 Jan 2023
+
+2
+NNs to OPF problems, and their results have higher accuracy
+than conventional NNs.
+The above-mentioned studies have made significant progress
+in designing elaborate network structures and loss functions.
+However, little attention has been paid to the training set
+generation problem. Specifically, they all adopt conventional
+probability sampling methods to produce datasets for training
+and testing, such as simple random sampling [10]–[13], [15],
+[17], [20], Monte Carlo simulation [18], or Latin hypercube
+sampling [16], [19]. These probability sampling methods can-
+not provide a theoretical guarantee that a generated training
+set can represent the input space of the OPF problem prop-
+erly. As a result, probability sampling methods may generate
+insufficient and unrepresentative training sets, so the trained
+NN solvers may provide unreliable solutions.
+It is important to create a sufficiently representative dataset
+for training an NN OPF solver. A training set’s representative-
+ness depends on its size and distribution in its feasible region
+[21]. Taking a medium-scale OPF problem as an example,
+millions of data samples may still be sparse given the high
+dimension of the NN’s inputs (e.g., operational parameters of
+the OPF problem: renewable generation and power demands
+at all buses); in addition, because the OPF problem is non-
+convex, the feasible region of the NN’s inputs is a complicated
+irregular space. Thus, generating a representative training
+set to cover all the feasible regions of the inputs with an
+acceptable size is quite challenging. Without a representative
+training set, it is difficult to guarantee that the NN OPF solver’s
+outputs are reliable, especially given “unseen” inputs in the
+inference process, as discussed in [22], [23].
+To address the above challenge, this study proposes
+a physical-model-integrated deep NN method with worth-
+learning data generation to solve AC-OPF problems. To the
+best of our knowledge, this is the first study that has addressed
+the representativeness problem of the training dataset for NN
+OPF solvers. The major contributions of this study are twofold:
+1) A novel physical-model-integrated NN is designed for
+solving the AC-OPF problem. This NN is constructed by
+a conventional MLP integrating an OPF-model module,
+which outputs not only the optimal decision variables
+of the OPF problem but also the violation degree of
+constraints. By penalizing the latter in the loss function
+during training, the NN can generate more reliable
+decision variables.
+2) Based on the designed NN, a novel generation method
+for worth-learning training data is proposed, which can
+identify samples in the input feasible region that are
+not well generalized by the previous NN. By iteratively
+applying this method during the training process, the
+trained NN gradually generalizes to the whole feasible
+region. As a result, the generalization and reliability of
+the proposed NN solver can be significantly enhanced.
+Furthermore, comprehensive numerical experiments are con-
+ducted, which prove that the proposed method is effective in
+terms of both reliability and optimality for solving AC-OPF
+problems with high computational efficiency.
+The remainder of this article is organized as follows. Section
+II provides preliminary models and the motivations behind this
+Fig. 1. The 3-bus system.
+study. Section III introduces the proposed method. Section IV
+details the experiments. Section V concludes this paper.
+II. ANALYSIS OF APPROXIMATING OPF PROBLEMS BY NN
+A. AC-OPF problem
+The AC-OPF problem aims to determine the optimal power
+dispatch (usually for generators) given specific operating con-
+ditions of a power system, e.g., power loads and renewable
+generation. A typical AC-OPF model can be formulated as
+min
+V , SG C
+�
+SG�
+(1a)
+s.t.:
+[V ] Y∗
+busV ∗ = SG − SL,
+(1b)
+SG ≤ SG ≤ S
+G,
+(1c)
+V ≤ V ≤ V,
+(1d)
+|YbV | ≤ ¯I,
+(1e)
+where Eq. (1a) is the objective, e.g., minimizing total genera-
+tion costs, and Eqs. (1b) to (1e) denote constraints. Symbols
+SG and SL are n × 1 vectors representing complex bus
+injections from generators and loads, respectively, where n
+is the number of buses. Symbol V is an n×1 vector denoting
+node voltages. Symbol [.] denotes an operator that transforms
+a vector into a diagonal matrix with the vector elements on
+the diagonal. Symbol Ybus is a complex n × n bus admittance
+matrix written as Y at other sections for convenience. Symbol
+Yb is a complex nb × n branch admittance matrix, and nb is
+the number of branches. The upper and lower bounds of any
+variable x are represented by ¯x and x, respectively. Vector ¯I
+denotes the current flow limit of branches.
+B. AC-OPF mapping from loads to optimal dispatch
+An NN model describes an input-output mapping. Specifi-
+cally, for an NN model solving the AC-OPF problem shown in
+Eq. (1), the input is the power demand SL, and the output is the
+optimal generation SG *. Hence, an NN OPF solver describes
+the mapping SG * = f OPF(SL). A well-trained NN should be
+able to accurately approximate this mapping.
+We provide a basic example of a 3-bus system, as shown in
+Fig. 1, to illustrate how NN works for OPF problems and
+explain the corresponding challenge for generalization. For
+simplicity, we assume there is no reactive power in the system
+and set r31 = r12 = 0.01 ; P i = 0 , P i = 4, for i ∈ {1, 3},
+P2 ∈ [−7, 0]; V i = 0.95 and V i = 1.05, for i ∈ {1, 2, 3}.
+
+Load
+P2E[-7,0]3
+Fig. 2. Examples of an NN fitting the OPF of the 3-bus system based on (a)
+simple random sampling, and (b) worth-learning data generation.
+Then, the OPF model Eq. (1) is reduced to the following
+quadratic programming:
+min
+V, PG P1 + 1.5 × P3
+(2a)
+s.t.: P1 = V1 (V1 − V2) /0.01 + V1 (V1 − V3) /0.01,
+(2b)
+P2 = V2 (V2 − V1) /0.01,
+(2c)
+P3 = V3 (V3 − V1) /0.01,
+(2d)
+0.95 ≤ V3 ≤ 1.05, 0.95 ≤ V2 ≤ 1.05,
+(2e)
+0 ≤ P1 ≤ 4, 0 ≤ P3 ≤ 4, V1 = 1,
+(2f)
+where V is [V1 V2 V3]⊤, and P G is [P1 P3]⊤.
+Given that P2 ranges from -7 to 0, the 3-bus OPF model can
+be solved analytically. The closed-form solution of [P ∗
+1 P ∗
+3 ] =
+f OPF
+3-bus(P2), is formulated as follows:
+P ∗
+1 =
+�
+50 − 50√0.04P2 + 1,
+c1,
+4,
+c2,
+(3)
+P ∗
+3 =
+�
+�
+�
+0,
+c1,
+213
+�
+1 − 0.34√0.04P2 + 1
+�2
++50√0.04P2 + 1 − 146
+,
+c2,
+(4)
+where c1 denotes condition 1:
+−3.84 ≤ P2 ≤ 0, and c2
+denotes condition 2: −7 ≤ P2 < −3.84.
+To further analyze the mapping f OPF
+3-bus, we draw the
+[P ∗
+1 P ∗
+3 ]–P2 curve according to Eqs. (3) and (4), shown in
+Fig. 2. Both the P ∗
+1 –P2 and P ∗
+3 –P2 curves are piecewise
+nonlinear functions, in which two oblique lines are nonlinear
+because of the quadratic equality constraints. The reason
+why the two curves above are piecewise is that the active
+inequalities change the [P ∗
+1 P ∗
+3 ]–P2 relationship. From an
+optimization perspective, each active inequality will add a
+unique equality constraint to the relationship, so the pieces
+in f OPF
+3-bus are determined by the sets of active inequalities. In
+this example, the two pieces in each curve correspond to two
+sets of active inequalities: P1 ≤ 4 and 0 ≤ P3. Moreover,
+the two intersection points are the critical points where these
+inequalities are just satisfied as equalities.
+For a general AC-OPF problem, its input is usually high-
+dimensional (commonly determined by the number of buses),
+and its feasible space is partitioned into some distinct regions
+by different sets of active inequality constraints. From an
+optimization perspective, a set of active constraints uniquely
+characterizes the relationship SG *
+= f OPF(SL), and the
+number of pieces theoretically increases with the number of
+inequality constraints by exponential order [24]–[26]. There-
+fore, there are massive regions, and each region corresponds
+to a unique mapping relation, i.e., a piece of mapping function
+f OPF.
+C. Challenges of fitting OPF mapping by NN
+As shown in Fig. 2(a), to fit the two-dimensional piecewise
+nonlinear curve of f OPF
+3-bus, we first adopt four data samples by
+simple random sampling and then use an NN to learn the
+curve. Obviously, there are significant fitting errors between
+the fitting and the original lines. Because the training set lacks
+the samples near the intersections in the curve (where p2 =
+−0.384 in this case), the NN cannot accurately approximate
+the mapping in the neighboring region of the intersections.
+A training set representing the whole input space is a prereq-
+uisite for an NN approximating the curve properly. However,
+it is nontrivial to generate a representative training set by
+probability sampling. As shown in Fig. 2(a), the intersections
+of f OPF are key points for the representativeness, and the
+number of intersections increases exponentially with that of
+the inequality constraints, as analyzed in II-B. When each
+sample is selected with a small possibility ρ, the generation
+of a dataset containing all the intersection points are in a low
+possibility event whose probability is equal to ρm, where m is
+the number of intersections. In practice, the only way to collect
+sufficient data representing the input space by probability
+sampling is to expand the dataset as much as possible [27].
+This is impractical for large power networks. Therefore, the
+conventional probability sampling in the literature can hardly
+produce a representative dataset with a moderate size.
+As shown in Fig. 2(b), if we are able to identify the two
+intersections, i.e., (P2 = −0.384, P1 = 4) and (P2 = −0.384,
+P3 = 0), and include them as new samples in the training
+dataset, the corresponding large fitting errors of the NN
+can be eliminated. These samples are termed as the worth-
+learning data samples. The focus of this study is to propose a
+worth-learning data generation method that can help identify
+worth-learning data samples and overcome the aforementioned
+disadvantage of conventional probability sampling (detailed in
+the following section).
+III. A PHYSICAL-MODEL-INTEGRATED NN WITH
+WORTH-LEARNING DATA GENERATION
+This section proposes a physical-model-integrated NN with
+a worth-learning data generation method to solve AC-OPF
+problems. The proposed NN is a combination of a fully-
+connected network and a transformed OPF model. It outputs
+not only the optimal decision variables of the OPF problem but
+also the violation degree of constraints, which provides guid-
+ance for identifying worth-learning data. The worth-learning
+data generation method creates representative training sets to
+enhance the generalization of the NN solver.
+
+P*
+Data from simple random sampling
+Fitting line
+Fitting error  Worth-learning data4
+START
+Initialize a training set by randomly sampling
+Train the NN on the current training set
+Identify worth-learning data 
+for the current NN
+Worth-learning data 
+are identified?
+Output the current NN
+END
+Y
+N
+Add identified 
+data to the 
+training set
+Fig. 3. Framework of the proposed training process.
+A. Framework of the proposed method
+The proposed data generation method has an iterative pro-
+cess, as shown in Fig. 3. First, a training set is initialized by
+random sampling; second, the physical-model-integrated NN
+is trained on the training set, where an elaborate loss function
+is utilized; third, worth-learning data for the current NN are
+identified; fourth, if worth-learning data are identified, these
+data are added to the training set and returns to the second
+step; otherwise, the current NN is output.
+The above training process converges until no worth-
+learning data are identified. This means that the training set
+is sufficiently representative of the input space of the OPF
+problem. As a result, the NN trained based on this dataset
+can generalize to the input feasible set well. The following
+subsections introduce the proposed method in detail.
+B. Physical-model-integrated NN
+In the second step of the proposed method (Fig. 3), the NN
+is trained to fit the mapping SG * = f OPF(SL). To obtain better
+results, we design a physical-model-integrated NN structure
+consisting of a conventional NN module and a physical-model
+module, as shown in Fig. 4. The former is a conventional MLP,
+while the latter is a computational graph transformed from the
+OPF model.
+1) Conventional NN module: This module first adopts a
+conventional MLP with learnable parameters to fit the mapping
+from the SL to the optimal decision variable V NN [28]. The
+V NN has its box constraint defined in Eq. (1). To ensure that
+the output V NN satisfies this constraint, we design a function
+dRe() to adjust any infeasible output V NN into its feasible
+region, which is formulated as follows:
+x ← dRe(x, x, x) = ReLU(x − x) − ReLU(x − x) + x,
+(5)
+where ReLU(x) = max(x, 0); x is the input of the function,
+and its lower and upper bounds are x and x, respectively. The
+diagram of this function is illustrated in Fig. 5.
+Input
+…
+…
+Physical model module
+Conventional NN module
+Output
+Fig. 4. The physical-model-integrated NN.
+Applying dRe() as the activation function of the last layer
+of the conventional MLP, the mathematical model of this
+conventional NN module is formulated as follows:
+V NN = MLP(SL),
+(6)
+V NN ← dRe(V NN, V , V ),
+(7)
+where Eq. (6) describes the conventional model of MLP and
+Eq. (7) adjusts the output of the MLP.
+2) Physical model module: This module receives V NN
+from the previous module, and then it outputs the optimal
+power generation SG
+phm and the corresponding constraints
+violation V iophm, where the subscript “phm” denotes the
+physical model module. The first output SG
+phm is the optimal
+decision variable of the AC-OPF problem. It can be calculated
+by V NN and SL, as follows:
+SG
+phm = [V NN]Y∗V ∗
+NN + SL.
+(8)
+The second output V iophm (termed as violation degree)
+measures the quality of SG
+phm and is the key metric to guide
+the proposed worth-learning data generation (see details in
+the following subsection III-C). Given V NN and SG
+phm, the
+violations of inequality constraints of the AC-OPF problem
+V iophm are calculated as follows:
+V ioS
+phm = ReLU(SG
+phm − S
+G) + ReLU(SG − SG
+phm),
+(9a)
+V ioI
+phm = ReLU(|YfV NN| − ¯I),
+(9b)
+V iophm = (V ioS
+phm
+V ioI
+phm)⊤,
+(9c)
+where V ioS
+phm denotes the violation of the upper or lower
+limit of SG
+phm, and V ioI
+phm represents the violation of branch
+currents.
+Remark 1. The physical-model-integrated NN is formed
+by combining the conventional NN module and the physical
+model module. It inputs SL and outputs SG
+phm and V iophm,
+as shown in Fig. 4. Its function is the same as conventional
+OPF numerical solvers. In addition, it is convenient for users
+Fig. 5. The dRe() function.
+
+5
+Feasible region
+Label value
+Region with tiny 
+predicted error
+Predicted value
+Effect of the proposed 
+loss function
+Point 1
+Point 2
+Fig. 6. Illustration of the effectiveness of the three terms in the loss function.
+to directly determine whether the result of the NN OPF solver
+is acceptable or not based on the violation degree V iophm. In
+contrast, most NN OPF solvers in the literature are incapable
+of outputting the violation degree directly [10]–[12].
+3) Loss function: To enhance the training accuracy of the
+physical-model-integrated NN, we design an elaborate loss
+function, which consists of V NN from the conventional NN
+module, and SG
+phm and V iophm from the physical model
+module. The formula is as follows:
+loss = || ˆV − V NN||1 + || ˆ
+SG − SG
+phm||1 + V iophm,
+(10)
+where ˆV and ˆ
+SG are label values from the training set, which
+is a ground truth dataset from numerical solvers.
+Combining the three terms in the loss function can help en-
+hance fitting precision. As shown in Fig. 6, if the loss function
+only has the first two items || ˆV − V NN||1+ || ˆ
+SG − SG
+phm||1 to
+penalize conventional fitting errors, the predicted value will be
+in a tiny square space (the red square in Fig. 6) around the
+label value. From the optimization perspective, the optimal
+label value is usually on the edge of its feasible region (the
+blue polyhedron in Fig. 6). This edge through the label value
+splits the square into two parts: the feasible (blue) part and
+the infeasible (white) part. Intuitively, we would prefer the
+predicted values to be in the feasible part. Thus, we also
+penalize violation degree V iophm in the loss function to force
+the predicted values with big V iophm close to the square’s
+feasible half space for smaller constraint violations.
+Although the proposed NN with elaborate loss function has
+high training accuracy, it is still difficult to guarantee the gen-
+eralization of the NN OPF solver to the whole input space with
+conventional random sampling. Therefore, it is indispensable
+and challenging to obtain a representative training dataset with
+moderate size to train the proposed NN, which is the focus of
+the following subsection.
+C. Worth-learning data generation
+As shown in Fig. 3, we adopt an iterative process to identify
+the worth-learning data. For an NN trained in the previous
+iteration, we utilize its output V iophm to help identify new
+data samples that are not yet properly generalized. Specifically,
+if an input SL* is feasible for the original OPF problem while
+the current NN outputs a large violation degree V io∗
+phm, the
+contradiction means the NN has a large fitting error at SL*.
+Input feasible set module
+Fig. 7. The input feasible set module.
+This is probably because sample SL* was not included in
+the previous training set and was not generalized by the NN.
+Hence, this sample SL* can be regarded as a worth-learning
+sample. Including the sample in the training dataset in the next
+iteration helps enhance the generalization of the NN.
+The key to the proposed worth-learning data generation
+method is to identify worth-learning samples efficiently. In-
+stead of traversing all of the possible inputs, we maximize
+V iophm for a given NN to identify the input with a large vio-
+lation degree. However, the inputs identified in the maximizing
+process should be feasible for the original OPF problem.
+Otherwise, the found inputs might be infeasible and useless
+for the representation of the training data.
+1) Input feasible set module: To keep the inputs identified
+in the maximizing process feasible for the original OPF
+problem, we formulate the input feasible set module to restrict
+power loads SL to their feasible set. The feasible set is
+composed of box constraints, current limits, and KCL&KVL
+constraints, which are transformed from the feasible set of the
+OPF problem defined in Eq. (1). The partial formulations of
+the input feasible set are as follows, where the subscript “ifs”
+denotes the input feasible set module:
+SG
+ifs = dRe
+�
+S′G
+ifs, SG, S
+G�
+, S′G
+ifs ∈ Rn,
+(11a)
+V ifs = dRe
+�
+V ′
+ifs, V , V
+�
+, V ′
+ifs ∈ Rn,
+(11b)
+SL
+ifs = SG
+ifs − [V ifs]Y∗V ∗
+ifs,
+(11c)
+Iifs = YbV ifs,
+(11d)
+where S′G
+ifs and V ′ifs are auxiliary n × 1 vectors in Rn and
+have no physical meaning. Symbols SG
+ifs and V ifs are restricted
+in their box constraints in Eqs. (11a) and (11b). Then the
+KCL&KVL correlations of SL
+ifs, SG
+ifs, and V ifs are described
+by Eq. (11c). Symbol Iifs in Eq. (11d) denotes the currents at
+all branches.
+The other formulations of the input feasible set aim to calcu-
+late V ioifs, the AC-OPF’s constraint violations corresponding
+to SL
+ifs and Iifs, as follows:
+V ioS
+ifs = ReLU(SL
+ifs − S
+L) + ReLU(SL − SL
+ifs),
+(12a)
+V ioI
+ifs = ReLU(|Iifs| − I),
+(12b)
+V ioifs = (V ioS
+ifs
+V ioI
+ifs)⊤,
+(12c)
+where V ioS
+ifs denotes the violation of the upper or lower limit
+of SL
+phm, and V ioI
+ifs denotes the violation of branch current.
+Remark 2. This module takes S′G
+ifs and V ′ifs as the inputs,
+and then outputs SL
+ifs and V ioifs, as shown in Fig. 7. When
+V ioifs = 0, the corresponding SL
+ifs lies in the feasible set of
+
+6
+Conventional 
+NN module
+Physical 
+model module
+Input 
+feasible set 
+module
+Updated 
+variables
+Fig. 8.
+The novel NN for max violation backpropagation by integrating
+physical-model-integrated NN with the input feasible set module.
+the AC-OPF problem. To identify feasible SL
+ifs in the process of
+maximizing V iophm, this module backpropagate the ∂V iophm
+∂SL
+ifs
+with V ioifs ≤ ζ (ζ is a small positive tolerance), and then
+it updates S′G
+ifs and V ′ifs. As a result, the corresponding SL
+ifs
+is always feasible. Furthermore, because S′G
+ifs and V ′ifs are
+not bounded, changing them can theoretically find any feasible
+SL
+ifs.
+2) Max violation backpropagation:
+To identify worth-
+learning data, a novel NN is created by inputting SL
+ifs into
+the physical-model-integrated NN (see Fig. 8). This NN has
+two outputs, i.e., V iophm and V ioifs. The former measures
+the constraint violation degree of the OPF solution SG*; the
+latter indicates the feasibility of the OPF input SL
+ifs. If SL
+ifs
+is a feasible input, i.e., V ioifs ≤ ζ, but the optimal solution
+SG* is infeasible, i.e., V iophm ≥ ξ (ξ is a threshold), this
+means the corresponding input is worth learning (i.e., it is
+not learned or generalized by the current NN). Based on this
+analysis, we design the loss function lossmax for max violation
+backpropagation, as follows:
+lossmax = V iophm − λ × V ioifs,
+(13)
+where λ is a large, constant weight parameter. When maxi-
+mizing this loss function, the algorithm tends to find a worth-
+learning SL
+ifs that has small V ioifs but large V iophm.
+During the max violation backpropagation, the proposed
+algorithm maximizes lossmax to update the variables S′G
+ifs
+and V ′ifs by gradient backpropagation until lossmax converges
+to the local maximum. After the process, the corresponding
+SL
+ifs is also found. Because the maximizing process can be
+processed in parallel by the deep learning module PyTorch,
+the worth-learning samples are found in batch, where the
+max violation backpropagation uses the previous training set
+as initial points to identify the new data. Further, the auto-
+differentiation technique in PyTorch can accelerate the process
+of parallel computation. Based on these techniques, massive
+worth-learning data samples are identified efficiently.
+D. Overall training process
+The overall training process is presented in Algorithm 1,
+which first takes an initial training dataset Dt (obtained by any
+conventional sampling method) as input. The learning rate η is
+equal to 10−3, the loss difference tolerance ϵ is equal to 10−2,
+the added dataset A is empty, and the loss difference ∆L is
+equal to infinity at initialization. The training is performed for
+a fixed number of epochs (lines 2–5). Then the max violation
+backpropagation starts to identify worth-learning data (lines
+6 and 7) by using the training data as the initial points (line
+8) and updating S′G
+ifs and V ′ifs until ∆L is less than ϵ (lines
+9–12), which indicates lossmax has converged to the terminal.
+Algorithm
+1
+Training
+process
+of
+the
+physical-model-
+integrated NN OPF solver with worth-learning data generation.
+Input: Dt =
+� ˆ
+SL, ˆV , ˆ
+SG�
+Initialization : η ← 10−3, ϵ ← 10−2, A ← ∅, ∆L ← ∞
+1: repeat
+2:
+for epoch k = 0, 1, ... do
+3:
+Train the NN with loss Eq. (10):
+4:
+w ← w − η∇loss.
+5:
+end for
+6:
+while ∆L ≥ ϵ do
+7:
+Identify data with lossmax Eq. (13):
+8:
+S′G
+ifs, V ′
+ifs ← SG
+ifs, V ifs ← ˆ
+SG, ˆV
+9:
+S′G
+ifs ← S′G
+ifs + η∇lossmax
+10:
+V ′
+ifs ← V ′
+ifs + η∇lossmax
+11:
+∆L ← | lossmax,i − lossmax,i−100 |
+12:
+end while
+13:
+{V iophm,N} ← ffilter(V iophm,N ≥ ξ)
+14:
+Collect {SL
+ifs} corresponding to {V iophm,N} based on the
+novel NN in Fig. 8
+15:
+Calculate { ˆV , ˆ
+SG} corresponding to {SG
+ifs} using numerical
+solvers
+16:
+A ← {SL
+ifs, ˆV , ˆ
+SG}
+17:
+Dt ← Dt ∪ A
+18: until A is ∅
+After the max violation backpropagation, a series of com-
+mands are designed to add proper data to the training set.
+First, a filter function ffilter is employed to eliminate data with
+terminal violation V iophm,N less than a given threshold ξ (the
+value depends on the acceptable violation settings). Second,
+{ ˆV , ˆ
+SG} is calculated by numerical solvers corresponding to
+SL
+ifs with large violation degree (lines 14 and 15). They consist
+of added set A (line 16). Third, the training set Dt is expanded
+with A (line 17). The loop is repeated until the added set A is
+empty (line 18), meaning no worth-learning data are identified.
+E. Efficiency and convergence of the proposed method
+Unlike general training processes for conventional NNs, the
+proposed physical-model-integrated NN with worth-learning
+data generation adopts an iterative training process. It iter-
+atively checks the NN’s generalization to the input’s feasi-
+ble space by identifying worth-learning data, as shown in
+Fig. 3 and Algorithm 1. This difference introduces two critical
+questions. 1) Efficiency: is the process of identifying worth-
+learning data computationally efficient? 2) Convergence: is
+the training set representative of the whole input space after
+iterations? In terms of the computational efficiency of the
+proposed method, the theoretical analysis (detailed in the
+Appendix A) shows it takes no more than 0.08 s to find
+one sample, which brings little computational burden into
+the training process. According to the experiment results, the
+average consumption time for finding one sample is 0.056 s. In
+terms of the convergence, we prove that the training set would
+gradually represent the whole input space in the Appendix
+B, because the number of worth-learning samples identified
+would converge to zero after a finite number of iterations.
+
+7
+The number of times the sequence codes are 
+repeated in the data generation loop (/100)
+The violation degree (MW)
+Fig. 9. Time consumption of the worth-learning data codes in three different
+iterations. The number of times the sequence codes are repeated in the data
+generation loop (x-axis) represents the time consumed in one data generation
+loop; the violation degrees (y-axis) quickly converge to the terminal stage.
+IV. NUMERICAL EXPERIMENTS
+The proposed method is evaluated using the IEEE 12-bus,
+14-bus, 30-bus, 57-bus, and 118-bus systems. The ground truth
+datasets are constructed using PANDAPOWER based on a
+prime-dual interior points algorithm.
+A. The efficiency of worth-learning data generation
+As shown in Algorithm 1, the proposed worth-learning data
+generation (lines 6–12) is the second loop in one iteration
+(lines 1–18), and the number of initial data points for the
+generation varies with iterations (lines 8, 15–17). To evalu-
+ate the efficiency of the worth-learning data generation, we
+conduct an experiment on the IEEE 57-bus system in three
+different iterations to quantitatively measure how much time
+it takes to finish one worth-learning data generation loop. The
+time consumption of the data-generation loops in the three
+different iterations is illustrated in Fig. 9. The x-axis is the
+number of times the codes are repeated (lines 6–12) divided
+by 100, which represents the time consumed in one data
+generation loop; the y-axis is the violation degree. The three
+lines converge to the terminal stage within 4000 times. The
+trends are similar: they increase very quickly at first (with 100
+epochs) and then approach the local maximum slowly (with
+2900–3900 epochs). The inflection points on the three lines
+are (1, 7228), (1, 9065), and (1, 5841).
+In the three iterations, 300, 500, and 800 new data samples
+are identified. Each data-generation loop in iterations takes
+30 s on average to run 3000–4000 times. Hence, one worth-
+learning data sample costs (30×3)/(300+500+800) ≈ 0.056
+s, which introduces little computational burden into the train-
+ing process compared to the other steps in Algorithm 1. For ex-
+ample, each label value calculated by numerical solvers costs
+around 1 s (line 14), and the NN training on a dataset with
+1100 samples costs around 600 s (lines 2–5). In conclusion,
+the numerical experiment verifies that the worth-learning data
+generation brings little computational burden to the training
+process.
+Furthermore, we list the time consumption comparison of
+the conventional and proposed training processes in Table I,
+where the conventional training process uses simple random
+sampling in place of the data generation loop (lines 6–12)
+in Algorithm 1. By comparing the time consumption of the
+TABLE I
+TRAINING TIME BASED ON THE CONVENTIONAL SIMPLE RANDOM
+SAMPLING AND PROPOSED WORTH-LEARNING DATA GENERATION
+Cases
+Conventional (min.)
+Proposed (min.)
+30-bus
+27.9
+30.1
+57-bus
+79.8
+85.5
+118-bus
+174.1
+181.2
+two methods, we can conclude that the training time of the
+proposed method only increases by 4%–8%. Hence, these
+experiments validate that the proposed worth-learning data
+generation is computationally efficient.
+B. Reliability and optimality of the proposed solver
+To validate the superiority of the proposed NN OPF solver
+(denoted by Proposed NN), we compare it with two bench-
+marks: 1) B1 NN, which adopts the conventional loss function
+and NN model (MLP) with a training dataset generated
+by simple random sampling; 2) B2 NN, which adopts the
+proposed loss function and physical-model-integrated model
+with a training dataset generated by simple random sampling.
+A particular test set different from the training datasets
+above is created to examine the effect of these models fairly.
+The test set has 600 samples that are produced by uniformly
+sampling 200 points in [80%, 120%] of the nominal value
+of one load three times. The other loads in the three times
+are fixed at light (80% × nominal value), nominal (100% ×
+nominal value), and heavy (120%×nominal value) load con-
+ditions. The load sampled has the largest nominal value to
+cover a big region of the input space. Based on these settings,
+the test set includes much “unseen” data for those models.
+The reliability of the NN OPF solvers is evaluated by the
+constraint violation degrees on all test data. The optimality loss
+is evaluated by the relative error between predicted results and
+label values. For a fair comparison, the three methods all stop
+their training processes when the value of || ˆV − V NN||1 is
+less than 2 × 10−4. In view of the iterative training process,
+the performance of the three solvers is studied with increasing
+training data, and the initial NNs are identical because they
+are trained on an initial dataset with N samples.
+The results are statistically analyzed by creating box plots
+displayed in Fig. 10. The violation degrees and optimality
+losses of the results of the NNs from the three methods con-
+verge to the terminal stages gradually. The rate of convergence
+of Proposed NN is the largest, that of B2 NN is in the middle,
+and that of B1 NN is the smallest.
+In Figs. 10(a) to 10(c), the comparison of the last violation
+degree gives notable results in the three cases. Specifically,
+the median values in three cases are 7, 15, and 75 for B1
+NN; 6, 12.5, and 60 for B2 NN; and 3.2, 6.1, and 25 for
+Proposed NN, respectively. The novel loss function brings a
+19% reduction of violation degree on average by comparing
+B1 NN and B2 NN. The proposed training data generation
+method introduces a 50% reduction of violation degree on
+average according to the comparison of B2 NN and Proposed
+NN. Moreover, the height of the last boxes in each subfigure
+suggests the robustness of the three solvers, and Proposed NN
+
+10000
+8000
+6000
+4000
+Data at 1st iteration
+2000
+Data at 2nd iteration
+Data at 3rd iteration
+10
+20
+30
+40
+0
+The number of epoches （/1008
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Proposed NN
+B1 NN
+B2 NN
+Fig. 10. The violation degree and optimality loss of the results of the NNs
+trained by three methods change with the number of training data in different
+cases: (a), (d) IEEE 30-bus; (b), (e) IEEE 57-bus; (c), (f) IEEE 118-bus.
+has the smallest height in all three cases, which indicates the
+worth-learning data generation can improve the reliability in
+encountering “unseen” data from the feasible region.
+The comparison of optimality losses is similar to that of
+violation degrees, as illustrated in Figs. 10(d) to 10(f). The
+proposed NN method has the best results in the three cases,
+and the final median values of optimality losses are 0.6%,
+0.5%, and 0.3% in the three different cases, respectively. The
+optimality losses of B2 NN and B1 NN increase by 150%,
+66%, and 360% and 142%, 167%, and 460% compared to
+those of the proposed NN method in the three cases.
+In conclusion, the proposed physical-model-integrated NN
+OPF solver with worth-learning data generation can improve
+the generalization of NN models compared to the conventional
+NN solvers. Specifically, the proposed method introduces an
+over 50% reduction of constraint violations and optimality
+losses in the results on average.
+C. Comparison with numerical solvers
+To further evaluate the capability of the proposed method,
+the next experiment focuses on the comparison with the
+results of the classical AC-OPF solver based on the prime-
+dual interior points algorithm and the classical DC-OPF solver
+with a linear approximation of the power flow equations. The
+classical AC-OPF solver produces the optimal solutions as
+the ground truth values, and the DC-OPF solver is a widely
+used approximation in the power industry. The test set is the
+same as that in Section IV-B. The performance of the three
+methods is evaluated by the following metrics: 1) the average
+consumption time to solve an OPF problem; 2) the average
+constraint violation degree V iophm, which is calculated by
+Eqs. (8) and (9) for the two numerical solvers; and 3) the
+average relative error of dispatch costs. These three metrics are
+denoted as Time (ms), Vio.(MW), and Opt.(%), respectively.
+The results are tabulated in Table II. The bottom row of the
+table shows the average results over the three cases. As shown,
+the proposed method achieves high computational efficiency,
+which is at least three orders of magnitude faster than the
+DC-OPF solver and four orders of magnitude faster than the
+AC-OPF solver. Furthermore, the method also has much lower
+constraint violations and optimality losses compared with the
+DC OPF solver. The average Vio. (MW) and Opt. (%) of the
+proposed solver are only 10.882 and 0.462, which are 44%
+and 18% of those of the DC-OPF solver, respectively.
+D. Interpretation of worth-learning data generation
+This subsection interprets why the worth-learning data gen-
+erated by the proposed method improve the representativeness
+of the training dataset. The proposed worth-learning data
+generation method is compared with the conventional simple
+random sampling method. Without loss of generality, the
+experiment is conducted on the 14-bus system. Beginning
+with an identical initial dataset, the conventional and proposed
+methods generate 100 samples in every step, and there are 8
+steps for both. To visualize the representativeness, we draw the
+distribution of these high-dimensional training samples based
+on the t-distributed Stochastic Neighbor Embedding algorithm
+[29], [30], which is a statistical method for visualizing high-
+dimensional data by giving each data point a location in a two-
+or three-dimensional map.
+The reduced-dimensional data distributions of the conven-
+tional and proposed methods are shown in Fig.11. In Fig.
+11(a), the data are produced by the simple random sampling
+method, and their distribution is almost in a “�” region,
+which means the possibility of sampling in this region is high.
+Furthermore, the new data added in each step overlap with
+existing data or fill in the intervals. The new data overlapping
+with existing data are redundant in terms of NN training.
+The data filling in the intervals may be also redundant when
+the blanks are generalized well by the trained NN model. In
+contrast, as shown in Fig. 11(b), the new data generated by
+
+(MW)
+X101
+8
+10
+12
+16
+2
+The number of the training data (320 + x X 64)(MW
+12
+16
+The number of the training data (320 + x X 64)× 102
+(MW
+Vio.
+10
+12
+16
+The number of the training data (320 + x X 64)%
+Optimality loss(
+2
+8
+10
+12
+16
+3
+4
+6
+The number of the training data (320 + x X 64)%
+Optimality loss(
+2
+8
+10
+12
+3
+6
+16
+The number of the training data (320 + x X 64)（%）
+2
+3
+4
+8
+10
+12
+16
+6
+The number of the training data (320 + x X 64)9
+TABLE II
+PERFORMANCE COMPARISON OF NUMERICAL SOLVERS AND THE PROPOSED SOLVER
+Test
+cases
+AC-OPF solver
+DC-OPF solver
+Proposed NN solver
+Time (ms)
+Vio. (MW)
+Opt. (%)
+Time (ms)
+Vio. (MW)
+Opt. (%)
+Time (ms)
+Vio. (MW)
+Opt. (%)
+30-bus
+530.3
+0
+0
+14.8
+5.340
+0.908
+0.110
+4.415
+0.603
+57-bus
+991.6
+0
+0
+36.2
+15.611
+1.758
+0.113
+7.226
+0.499
+118-bus
+1606.7
+0
+0
+78.5
+52.199
+4.762
+0.116
+21.004
+0.285
+Avg.
+1024.9
+0
+0
+129.5
+24.383
+2.476
+0.113
+10.882
+0.462
+(a) Training dataset generated by simple random sampling
+(b) Training dataset generated by worth-learning data generation method
+Fig. 11. Reduced-dimensional distributions of the training datasets generated
+by two different methods.
+the proposed method in each step hardly overlap with existing
+data and are usually outside the region covered by the initial
+data. These new data increase the area covered by the training
+set so that the training set can have better representativeness
+of the input feasible region. This explains the effectiveness of
+the proposed worth-learning data generation method.
+V. CONCLUSION
+This study proposes an AC-OPF solver based on a physical-
+model-integrated NN with worth-learning data generation to
+produce reliable solutions efficiently. To the best of our knowl-
+edge, this is the first study that has addressed the generalization
+problem of NN OPF solvers regarding the representativeness
+of training datasets. The physical-model-integrated NN is
+designed by integrating an MLP and an OPF-model module.
+This specific structure outputs not only the optimal decision
+variables of the OPF problem but also the constraint violation
+degree. Based on this NN, the worth-learning data generation
+method can identify feasible training samples that are not well
+generalized by the NN. Accordingly, by iteratively applying
+this method and including the newly identified worth-learning
+data samples in the training set, the representativeness of the
+training set can be significantly enhanced.
+The theoretical analysis shows that the method brings little
+computational burden into the training process and can make
+the models generalize over the feasible region. Experimen-
+tal results show that the proposed method leads to over a
+50% reduction of both constraint violations and optimality
+loss compared to conventional NN solvers. Furthermore, the
+computation speed of the proposed method is three orders of
+magnitude faster than that of the DC-OPF solver.
+APPENDIX A
+COMPUTATIONAL EFFICIENCY OF WORTH-LEARNING
+DATA GENERATION
+To analyze the computational complexity of the proposed
+NN model with worth-learning data generation, we adopt a
+widely used measure—the number of floating-point operations
+(FLOPs) during the NN model’s forward-backward propaga-
+tion. The total FLOPs of one single layer of a fully-connected
+NN model can be calculated as follows:
+Forward :
+FLOPs = (2I − 1) × O,
+(14a)
+Backward :
+FLOPs = (2I − 1) × O,
+(14b)
+where I is the dimension of the layer’s input, and O is the
+dimension of its output.
+To approximate an OPF mapping based on a 57-bus system,
+the proposed NN model uses the following structure: 84 ×
+1000×2560×2560×5120×2000×114. According to Eq. (14),
+the total FLOPs of the NN per forward-backward process is
+around 1×108. The GPU used in the experiment is the Quadro
+P6000, and its performance is 12.2 TFLOP/s (1 TFLOP/s =
+1012 FLOP/s). Using the GPU, we can perform the forward-
+backward process 1.22 × 105 times per second.
+For the worth-learning data generation in Algorithm 1, the
+forward process is to calculate V ioifs and V iophm, and the
+backward process is to update S′G
+ifs and V ′ifs by the gradients.
+We concatenate S′G
+ifs and V ′ifs as a vector x, and we suppose
+the range of each item in x is [0, 10], and x changes 10−3
+in each update step. Varying from 0 to 10, it costs 104 times
+the forward-backward processes. In other words, the algorithm
+can at least update 1.22 × 105/104 ≈ 12 samples in 1 s, so
+finding one sample costs no longer than 0.08 s.
+In practice, there is a slight error between the actual speed
+in experiments and the theoretical analysis. According to the
+numerical experiments in Section IV-A, an average of 533
+samples are found in 30 s. The average consumption time for
+identifying one sample is 0.056 s.
+
+2nd step
+Initial data
+step
+80
+60
+40
+20
+D
+0
+-20
+-40
+-60
+-80
+-100
+8th step
+6th
+Overall data
+step
+80
+60
+40
+20
+D
+0
+-20
+-40
+-60
+-80
+-100
+-50
+50
+100
+-50
+-100
+0
+-100
+50
+100
+-100
+-50
+50
+100
+0
+D2
+D2
+D2Initial data
+2nd step
+4th step
+80
+60
+40
+20
+D
+0
+-20
+-40
+-60
+-80
+-100
+6th step
+8th step
+Overall data
+80
+60
+40
+20
+D
+0
+-20
+-40
+-60
+-80
+-100
+-100
+-50
+0
+50
+100
+-100
+-50
+0
+50
+100
+-100
+-50
+0
+50
+100
+D2
+D2
+D2
+Initial data
+Identified data in previous steps
+New data10
+Fig. 12.
+Illustration of the covered region Scover expanding its area by the
+generalized region Sadd.
+From the analysis presented above, we can conclude that the
+proposed worth-learning data generation method brings little
+computational burden into the training process.
+APPENDIX B
+CONVERGENCE OF WORTH-LEARNING DATA GENERATION
+This section verifies that the proposed NN with worth-
+learning data generation can generalize to the whole feasible
+set. NN models are continuous functions because both linear
+layers and activation functions are continuous. We define a
+critical violation value ϵ that divides the input space into
+two types: the covered region (the V iophm values of all of
+the points are less or equal to ϵ) and the uncovered region
+(the V iophm values of all of the points are greater than
+ϵ). The boundaries of the two regions consist of the points
+whose V iophm values are approximately equal to ϵ. Using
+these points as initial points, we can identify points with the
+local maximum in the uncovered region by max violation
+backpropagation.
+Next, these new points {x1} (the red points) are added to
+the training set. After training, the neighborhood of these new
+points {x1} would be covered. Due to the generalization of
+NNs, most points in the area Sadd = {x|a × xini
+0 + (1 − a) ×
+x1, 0 ≤ a ≤ 1} would also be covered, where {xini
+0 } are the
+initial points on the boundaries (the black points), as shown
+in Fig. 12.
+Therefore, the area Sadd is subtracted from the uncovered
+region. Through iterations, the uncovered region is emptied,
+and the number of added samples converges to zero.
+In practice, we choose the training set instead of the
+boundary points as initial points for convenience. Although
+some samples in the training set are not at boundaries, they
+are eliminated by the filter function, as shown in Algorithm 1.
+Therefore, the replacement of the boundary points has no
+impact on the results.
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+C

9tE2T4oBgHgl3EQfQQYW/content/tmp_files/2301.03767v1.pdf.txt

@@ -0,0 +1,1506 @@
+Online Backfilling with No Regret for Large-Scale Image Retrieval
+Seonguk Seo1,†
+Mustafa Gokhan Uzunbas3
+Bohyung Han1,2
+Sara Cao3
+Joena Zhang3
+Taipeng Tian3
+Ser-Nam Lim3
+1ECE & 1,2IPAI, Seoul National University
+3Meta AI
+{seonguk, bhhan}@snu.ac.kr
+{gokhanuzunbas, joenazhang, xuefeicao01, ttp, sernamlim}@meta.com
+Abstract
+Backfilling is the process of re-extracting all gallery em-
+beddings from upgraded models in image retrieval systems.
+It inevitably requires a prohibitively large amount of com-
+putational cost and even entails the downtime of the ser-
+vice. Although backward-compatible learning sidesteps this
+challenge by tackling query-side representations, this leads
+to suboptimal solutions in principle because gallery em-
+beddings cannot benefit from model upgrades. We address
+this dilemma by introducing an online backfilling algorithm,
+which enables us to achieve a progressive performance im-
+provement during the backfilling process while not sacri-
+ficing the final performance of new model after the com-
+pletion of backfilling. To this end, we first propose a sim-
+ple distance rank merge technique for online backfilling.
+Then, we incorporate a reverse transformation module for
+more effective and efficient merging, which is further en-
+hanced by adopting a metric-compatible contrastive learn-
+ing approach. These two components help to make the dis-
+tances of old and new models compatible, resulting in de-
+sirable merge results during backfilling with no extra com-
+putational overhead. Extensive experiments show the effec-
+tiveness of our framework on four standard benchmarks in
+various settings.
+1. Introduction
+Image retrieval models [5, 10, 21, 23] have achieved re-
+markable performance by adopting deep neural networks
+for representing images. Yet, all models need to be up-
+graded at times to take advantage of improvements in train-
+ing datasets, network architectures, and training techniques.
+This unavoidably leads to the need for re-extracting the fea-
+tures from millions or even billions of gallery images using
+the upgraded new model. This process, called backfilling
+† This work was mostly done during an internship at Meta AI.
+or re-indexing, needs to be completed before the retrieval
+system can benefit from the new model, which may take
+months in practice.
+To sidestep this bottleneck, several backfilling-free ap-
+proaches based on backward-compatible learning [4,13,19,
+20,22] have been proposed. They learn a new model while
+ensuring that its feature space is still compatible with the old
+one, thus avoiding the need for updating old gallery embed-
+dings. Although these approaches have achieved substantial
+performance gains without backfilling, they achieve feature
+compatibility at the expense of feature discriminability and
+their performance is suboptimal. We argue that backward-
+compatible learning is not a fundamental solution and back-
+filling is still essential to accomplish state-of-the-art perfor-
+mance without performance sacrifices.
+To resolve this compatibility-discriminability dilemma,
+we relax the backfill-free constraint and propose a novel
+online backfilling algorithm equipped with three technical
+components. We posit that an online backfilling technique
+needs to satisfy three essential conditions: 1) immediate de-
+ployment after the completion of model upgrade, 2) pro-
+gressive and non-trivial performance gains in the middle
+of backfilling, and 3) no degradation of final performance
+compared to offline backfilling. To this end, we first pro-
+pose a distance rank merge framework to make online back-
+filling feasible, which retrieves images from both the old
+and new galleries separately and merge their results to ob-
+tain the final retrieval outputs even when backfilling is still
+ongoing. While this approach provides a monotonic perfor-
+mance increase with the progress of backfilling regardless
+of the gallery of interest and network architectures, it re-
+quires feature computations twice, once from the old model
+and another from the new one at the inference stage of a
+query. To overcome this limitation, we introduce a reverse
+transformation module, which is a lightweight mapping net-
+work between the old and new embeddings. The reverse
+transformation module allows us to obtain the query repre-
+sentations compatible with both the old and new galleries
+arXiv:2301.03767v1  [cs.CV]  10 Jan 2023
+
+using only a single feature extraction. On the other hand,
+however, we notice that the scales of distance in the embed-
+ding spaces of the two models could be significantly dif-
+ferent. We resolve the limitation with a metric compatible
+learning technique, which calibrates the distances of two
+models via contrastive learning, further enhancing perfor-
+mance of rank merge.
+The main contributions of our work are summarized as
+follows.
+• We propose an online backfilling approach, a funda-
+mental solution for model upgrades in image retrieval
+systems, based on distance rank merge to overcome
+the compatibility-discriminability dilemma in existing
+compatible learning methods.
+• We incorporate a reverse query transform module to
+make it compatible with both the old and new galleries
+while computing the feature extraction of query only
+once in the middle of the backfilling process.
+• We adopt a metric-compatible learning technique to
+make the merge process robust by calibrating distances
+in the feature embedding spaces given by the old and
+new models.
+• The proposed approach outperforms all existing meth-
+ods by significant margins on four standard benchmark
+datasets under various scenarios.
+The rest of this paper is organized as follows. Section 2
+reviews the related works. We present the main framework
+of online backfilling in Section 3, and discuss the techni-
+cal components for improvement in Section 4 and 5. We
+demonstrate the effectiveness of the proposed framework in
+Section 6 and conclude this paper in Section 7.
+2. Related Work
+Backward compatible learning
+Backward compatibility
+refers to the property to support older versions in hardware
+or software systems. It has been recently used in model
+upgrade scenarios in image retrieval systems. Since the fea-
+ture spaces given by the models relying on training datasets
+in different regimes are not compatible [11, 24], model up-
+grades require re-extraction of all gallery images from new
+models, which takes a huge amount of computational cost.
+To prevent this time-consuming backfilling cost, backward
+compatible training (BCT) [1, 13, 15, 19, 22, 26] has been
+proposed to learn better feature representations while be-
+ing compatible with old embeddings, which makes the new
+model backfill-free. Shen et al. [19] employ the influence
+loss that utilizes the old classifier as a regularizer when
+training the new model. LCE [13] introduces an alignment
+loss to align the class centers between old and new mod-
+els and a boundary loss that restricts more compact intra-
+class distributions for the new model. Bai et al. [1] pro-
+pose a joint prototype transfer with structural regularization
+to align two embedding features. UniBCT [26] presents a
+structural prototype refinement algorithm that first refines
+noisy old features with graph transition and then conducts
+backward compatible training. Although these approaches
+improved compatible performance without backfilling, they
+clearly sacrifice feature discriminability to achieve feature
+compatibility with non-ideal old gallery embeddings.
+Compatible learning with backfilling
+To overcome the
+inherent limitation of backward compatible learning, sev-
+eral approaches [17, 20, 25] have been proposed to uti-
+lize backfilling but efficiently. Forward compatible train-
+ing (FCT) [17] learn a lightweight transformation mod-
+ule that updates old gallery embeddings to be compati-
+ble with new embeddings. Although it gives better com-
+patible performance than BCT, it requires an additional
+side-information [2] to map from old to new embeddings,
+which limits its practicality. Moreover, FCT still suffers
+from computational bottleneck until all old gallery embed-
+dings are transformed, especially when the side-information
+needs to be extracted.
+On the other hand, RACT [25]
+and BiCT [20] alleviate this bottleneck issue by backfilling
+the gallery embeddings in an online manner. RACT first
+trains a backward-compatible new model with regression-
+alleviating loss, then backfills the old gallery embeddings
+with the new model.
+Because the new feature space is
+compatible with the old one, the new model can be de-
+ployed right away while backfilling is carried out in the
+background.
+BiCT further reduces the backfilling cost
+by transforming the old gallery embeddings with forward-
+compatible training [17]. Although both approaches can
+utilize online backfilling, they still sacrifice the final perfor-
+mance because the final new embeddings are constrained by
+the old ones. Unlike these methods, our framework enables
+online backfilling while fully exploiting the final new model
+performance without any degradation.
+3. Image Retrieval by Rank Merge
+This section discusses our baseline image retrieval al-
+gorithm that makes online backfilling feasible.
+We first
+present our motivation and then describe technical details
+with empirical observations.
+3.1. Overview
+Our goal is to develop a fundamental solution via online
+backfilling to overcome the compatibility-discriminability
+trade-off in compatible model upgrade.
+This strategy
+removes inherent limitations of backfill-free backward-
+compatible learning—the inability to use state-of-the-
+
+Figure 1. Image retrieval with the proposed distance rank merge technique. In the middle of backfilling, we retrieve images independently
+using two separate models and their galleries, and merge the retrieval results based on their distances. Note that the total number of gallery
+embeddings are fixed throughout the backfilling process, i.e., |G| = |Gnew| + |Gold|.
+art representations of gallery images through model
+upgrades—while avoiding prohibitive costs, including the
+situation that we cannot benefit from model upgrade of the
+offline backfilling process, until backfilling is completed.
+To be specific, the proposed image retrieval system with
+online backfilling should satisfy the following three condi-
+tions:
+1. The system can be deployed immediately as soon as
+model upgrade is complete.
+2. The performance should monotonically increase with-
+out negative flips1 as backfill progresses.
+3. The final performance should not be sacrificed com-
+pared to the algorithm relying on offline backfilling.
+We present a distance rank merge approach for image re-
+trieval, which enables online backfilling in arbitrary model
+upgrade scenarios. Our method maintains two separate re-
+trieval pipelines corresponding to the old and new models
+and merges the retrieval results from the two models based
+on distances from a query embedding. This allows us to run
+the retrieval system without a warm-up period and achieve
+surprisingly good results during the backfill process. Note
+that the old and new models are not required to be com-
+patible at this moment but we will make them so to further
+improve performance in the subsequent sections.
+3.2. Formulation
+Let q ∈ Q be a query image and G = {g1, ..., gN} be
+a gallery composed of N images. An embedding network
+φ(·) projects an image onto a learned feature embedding
+space. To retrieve the closest gallery image given a query,
+we find arg ming∈G dist (φ(q), φ(g)), where dist(·, ·) is a
+distance metric. Following [19], we define the retrieval per-
+formance as
+M(φ(Q), φ(G)),
+(1)
+1The “negative flip” refers to performance degradation caused by in-
+correct retrievals of samples by the new model, which were correctly rec-
+ognized by the old model.
+where M(·, ·) is an evaluation metric such as mean aver-
+age precision (mAP) or cumulative matching characteristics
+(CMC), and φ(·) indicates embedding models for query and
+gallery, respectively.
+Backward compatibility
+Denote the old and new embed-
+ding networks by φold(·) and φnew(·) respectively. If φnew(·)
+is backward compatible with φold(·), then we can perform
+search on a set of old gallery embeddings using a new
+query embedding, i.e., arg ming∈G dist(φnew(q), φold(g)).
+As stated in [19], the backward compatibility is achieved
+when the following criterion is satisfied:
+M(φnew(Q), φold(G)) > M(φold(Q), φold(G)).
+(2)
+From now, we refer to a pair of embedding networks for
+query and gallery as a retrieval system, e.g., {φ(·), φ(·)}.
+Rank merge
+Assume that the first M out of a total of
+N images are backfilled, i.e., Gnew = {g1, ..., gM} and
+Gold = {gM+1, ..., gN}. Note that the total number of
+stored gallery embeddings is fixed to N during the back-
+filling process, i.e., Gold = G − Gnew. Then, we first con-
+duct image retrieval using the individual retrieval systems,
+{φold, φold} and {φnew, φnew}, independently as
+gm = arg min
+gi∈Gold dist
+�
+φold(q), φold(gi)
+�
+,
+(3)
+gn = arg min
+gj∈Gnew dist (φnew(q), φnew(gj)) .
+(4)
+Figure 1 illustrates the retrieval process. For each query
+image q, we finally select gm if dist(φold(q), φold(gm)) <
+dist(φnew(q), φnew(gn)) and gn otherwise.
+The retrieval
+performance after rank merge during backfilling is given by
+Mt :=
+(5)
+M({φold(Q), φnew(Q)}, {φold(Gold
+t ), φnew(Gnew
+t
+)}),
+where t ∈ [0, 1] indicates the rate of backfilling completion,
+i.e., |Gnew
+t
+| = t|G| and |Gold
+t | = (1 − t)|G|. The criteria
+
+Old retrieval system
+retrieval
+Plo
+dold (Gold)
+Query
+Gallery (G)
+(q)
+retrieval
+IG| = |Gnew| + |Gold]
+backfilling
+New retrieval system anew0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.62
+0.64
+0.66
+0.68
+0.70
+0.72
+0.74
+0.76
+mAP
+New (0.773)
+Merge (0.693)
+Old (0.627)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.84
+0.86
+0.88
+0.90
+CMC (Top1 Acc.)
+New (0.909)
+Merge (0.871)
+Old (0.827)
+(a) ImageNet-1K
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.25
+0.30
+0.35
+0.40
+0.45
+mAP
+New (0.474)
+Merge (0.308)
+Old (0.216)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+CMC (Top1 Acc.)
+New (0.626)
+Merge (0.490)
+Old (0.343)
+(b) CIFAR-100
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.17
+0.18
+0.19
+0.20
+0.21
+0.22
+0.23
+mAP
+New (0.234)
+Merge (0.195)
+Old (0.165)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.32
+0.34
+0.36
+0.38
+CMC (Top1 Acc.)
+New (0.391)
+Merge (0.358)
+Old (0.307)
+(c) Places-365
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.35
+0.40
+0.45
+0.50
+mAP
+New (0.513)
+Merge (0.400)
+Old (0.312)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.50
+0.55
+0.60
+0.65
+0.70
+CMC (Top1 Acc.)
+New (0.703)
+Merge (0.639)
+Old (0.497)
+(d) Market-1501
+Figure 2. mAP and CMC results on the standard benchmarks using ResNet-18. Old and New denote the performance without backfilling
+and with offline backfilling, respectively. The proposed distance rank merging of the old and new models, denoted by Merge, exhibits
+desirable results; the performance monotonically increases as backfill progresses without negative flips for all datasets and our algorithm
+based on online backfilling achieves competitive final performances with offline backfilling. The numbers in the legend indicate either
+AUCmAP or AUCCMC scores.
+discussed in Section 3.1 are formally defined as
+M0 ≥ M(φold(Q), φold(G)),
+(6)
+M1 ≥ M(φnew(Q), φnew(G)),
+(7)
+Mt1 ≥ Mt2 if t1 ≥ t2.
+(8)
+Comprehensive evaluation
+To measure both backfilling
+cost and model performance comprehensively during online
+backfilling, we utilize the following metrics that calculate
+the area under mAP or CMC curves as
+AUCmAP =
+� 1
+0
+mAPtdt and AUCCMC =
+� 1
+0
+CMCtdt.
+3.3. Merge Results
+We present the results from the rank merge strategy on
+two standard benchmarks, including ImageNet-1K [18] and
+Places-365 [28], in Figure 2. Our rank merging approach
+yields strong and robust results for all datasets; both mAP
+and CMC monotonically increase without negative flips as
+backfill progresses even though the old and new models are
+not compatible each other. Also, it takes full advantage of
+the new model until the end of backfilling without suffering
+from performance degradation. This validates that our rank
+merge technique satisfies the criteria for online backfilling
+discussed in Section 3.1 and 3.2. Please refer to Section 6.1
+for the experimental detail.
+Figure 3. Reverse query transform module, ψ(·), learns a mapping
+from new to old feature spaces. We only update the parameters of
+the module ψ(·) (in red rectangle) during training.
+4. Reverse Query Transform
+Our baseline image retrieval method is model-agnostic,
+free from extra training, and effective for performance im-
+provement. However, one may argue that the proposed ap-
+proach is computationally expensive at inference time be-
+cause we need to conduct feature extraction twice per query
+for both the old and new models. This section discusses how
+to alleviate this limitation by introducing a small network,
+called the reverse query transform module.
+4.1. Basic Formulation
+To reduce the computational cost incurred by comput-
+ing query embeddings twice at inference stage, we compute
+the embedding using the new model and transform it to the
+version compatible with the old model through the reverse
+
+old
+tnew
+(.)
+new
+revFigure 4. Image retrieval merging with reverse query transform module. Backward retrieval system consists of reversely transformed new
+query and old gallery, {φrev, φold}. The final image retrieval results are given by merging the outputs from {φrev, φold} and {φnew, φnew}.
+query transform module as illustrated in Figure 3. To estab-
+lish such a mechanism, we fix the parameters of the old and
+new models {φold, φnew} after training them independently,
+and train a lightweight network, ψ(·), which transforms the
+embedding in the new model to the one in the old model.
+For each training example x, our objective is minimizing
+the following loss:
+LRQT(x) := dist
+�
+ψ (φnew(x)) , φold(x)
+�
+,
+(9)
+where dist(·, ·) is a distance metric such as ℓ2 or cosine dis-
+tances. Because we only update the parameters in ψ(·), not
+the ones in φnew(·) or φold(·), we can still access the repre-
+sentations given by the new model at no cost even after the
+optimization of ψ(·). Note that this reverse query transform
+module differs from FCT [17] mainly in terms of transfor-
+mation direction and requirement of side information. FCT
+performs a transformation from the old representation to the
+new, while the opposite is true for our proposed approach.
+Since the embedding quality of a new model is highly likely
+to be better than that of an old one, our reverse transforma-
+tion module performs well even without additional side in-
+formation and, consequently, is more practical and efficient.
+4.2. Integration into Baseline Retrieval System
+Figure 4 illustrates the distance rank merge process to-
+gether with the proposed reverse transformation module.
+The whole procedure consists of two retrieval systems de-
+fined by a pair of query and gallery representations, back-
+ward retrieval system {φrev, φold} and new retrieval system
+{φnew, φnew}, where φrev := ψ(φnew). Note that we obtain
+both the new and compatible query embeddings, φnew(q)
+and φrev(q) = ψ(φnew(q)), using a shared feature extrac-
+tion network, φnew(·).
+The entire image retrieval pipeline consists of two parts:
+1) feature extraction of a query image and 2) search for the
+nearest image in a gallery from the query. Compared to the
+image retrieval based on a single model, the computational
+cost of the proposed model with rank merge requires negli-
+gible additional cost, which corresponds to feature transfor-
+mation ψ(·) in the first part. Note that the number of total
+gallery embeddings is fixed, i.e., |Gnew| + |Gold| = |G|, so
+the cost of the second part is always the same in both cases.
+5. Distance Calibration
+While the proposed rank merge technique with the ba-
+sic reverse transformation module works well, there ex-
+ists room for improvement in calibrating feature embedding
+spaces of both systems. This section discusses the issues in
+details and presents how we figure them out.
+5.1. Cross-Model Contrastive Learning
+The objective in (9) cares about the positive pairs φold
+and φrev with no consideration of negative pairs, which can
+sometimes lead to misranked position. To handle this issue,
+we employ a supervised contrastive learning loss [7, 14] to
+consider both positive and negative pairs as follows:
+LCL(xi, yi) = − log
+�
+yk=yi sold
+ik
+�
+yk=yi sold
+ik + �
+yk̸=yi sold
+ik
+,
+(10)
+where sold
+ij
+= exp
+�
+−dist
+�
+φrev(xi), φold(xj)
+��
+and yi de-
+notes the class membership of the ith sample. For more ro-
+bust contrastive training, we perform hard example mining
+for both the positive and negative pairs2. Such a contrastive
+learning approach facilitates distance calibration and im-
+proves feature discrimination because it promotes separa-
+tion of the positive and negative examples.
+Now, although the distances within the backward re-
+trieval system {φrev, φold} become more comparable, they
+are still not properly calibrated in terms of the distances
+in the new retrieval system {φnew, φnew}. Considering dis-
+tances in both retrieval systems jointly when we train the
+reverse transformation module, we can obtain more com-
+parable distances and consequently achieve more reliable
+rank merge results. From this perspective, we propose a
+2For each anchor, we select the half of the examples in each of positive
+and negative labels based on the distances from the anchor.
+
+Backward retrieval system
+pold (.)
+retrieval
+(.)
+grev(q)
+dold (Gold)
+Gallery
+(G)
+Query
+retrieval
+(q)
+(q)
+backfilling
+New retrieval system [@new
+,dnew1Figure 5. Illustration of cross-model contrastive learning loss with
+backward retrieval system {φold, φrev} and new retrieval system
+{φnew, φnew}.
+Two boxes with dotted lines corresponds to two
+terms in (11). For each retrieval system, the distances between
+positive pairs are learned to be both smaller than those of negative
+pairs in the two retrieval systems.
+cross-model contrastive learning loss as
+LCMCL(xi, yi) =
+(11)
+− log
+�
+yk=yi sold
+ik
+�
+yk=yi sold
+ik + �
+yk̸=yi sold
+ik + �
+yk̸=yi snew
+ik
+− log
+�
+yk=yi snew
+ik
+�
+yk=yi snew
+ik + �
+yk̸=yi snew
+ik + �
+yk̸=yi sold
+ik
+,
+where snew
+ij
+= exp(−dist
+�
+φnew(xi), φnew(xj)
+�
+) and sold
+ij =
+exp(−dist
+�
+φrev(xi), φold(xj)
+�
+).
+Figure 5 illustrates the
+concept of the loss function. The positive pairs from the
+backward retrieval system {φrev, φold} are trained to locate
+closer to the anchor than not only the negative pairs from
+the same system but also the ones from the new system
+{φnew, φnew}, and vice versa. We finally replace (9) with
+(11) for training the reverse transformation module. Com-
+pared to (10), additional heterogeneous negative terms in
+the denominator of (11) play a role as a regularizer to make
+the distances from one model directly comparable to those
+from other one, which is desirable for our rank merge strat-
+egy.
+5.2. Training New Feature Embedding
+Until now, we do not jointly train the reverse transfor-
+mation module ψ(·) and the new feature extraction module
+φnew(·) as illustrated in Figure 3. This hampers the compat-
+ibility between the backward and new retrieval systems be-
+cause the backward retrieval system {φrev, φold} is the only
+part to be optimized while the new system {φnew, φnew} is
+fixed. To provide more flexibility, we add another transfor-
+mation module ρ(·) on top of the new model as shown in
+Figure 6, where ρnew = ρ(φnew) and ρrev = ψ(ρ(φnew)). In
+this setting, we use ρnew as the final new model instead of
+φnew, and our rank merge process employs {ρrev, φold} and
+Figure 6.
+Compatible training with learnable new embedding.
+Compared to Figure 3, another transformation module ρ(·) is in-
+corporated on top of the new model to learn new embedding fa-
+vorable to our rank merging. The retrieval results are now merged
+from {ρrev, φold} and {ρnew, ρnew}.
+{ρnew, ρnew} eventually. This strategy helps to achieve a bet-
+ter compatibility by allowing both systems to be trainable.
+The final loss function to train the reverse transformation
+module has the identical form to LCMCL in (11) except for
+the definitions of snew
+ij
+and sold
+ij , which are given by
+snew
+ij
+= exp (−dist (ρnew(xi), ρnew(xj)))
+(12)
+sold
+ij = exp
+�
+−dist
+�
+ρrev(xi), φold(xj)
+��
+.
+(13)
+Note that this extension does not result in computational
+overhead at inference stage but yet improves the perfor-
+mance even further.
+6. Experiments
+We present our experiment setting, the performance of
+the proposed approach, and results from the analysis of al-
+gorithm characteristics.
+6.1. Dataset and Evaluation Protocol
+We employ four standard benchmarks, which includes
+ImageNet-1K [18],
+CIFAR-100 [9],
+Places-365 [28],
+Market-1501 [27]. As in previous works [17,19], we adopt
+the extended-class setting in model upgrade; the old model
+is trained with examples from a half of all classes while the
+new model is trained with all samples. For example, on the
+ImageNet-1K dataset, the old model is trained with the first
+500 classes and the new model is trained with the whole
+1,000 classes.
+Following the previous works [17, 20, 25], we measure
+mean average precision (mAP) and cumulative matching
+characteristics (CMC)3. We also report our comprehensive
+results in terms of AUCmAP and AUCCMC at 10 backfill time
+slices, i.e., t ∈ {0.0, 0.1, ..., 1.0} in (5).
+6.2. Implementation Details
+We employ ResNet-18 [6], ResNet-50 [6], and ViT-
+B/32 [3] as our backbone architectures for either old or new
+3CMC corresponds to top-k accuracy, and we report top-1 accuracy in
+all tables and graphs.
+
+old
+rev
+new
+D
+anchor
+positive
+negativeold
+(.)
+p()
+()
+new
+rev
+0Table 1. Comparison with existing compatible learning methods on four standard benchmarks in homogeneous model upgrades. Gain
+denotes relative gain that each method achieves from old model in terms of AUCmAP, compared to the gain of new model. The proposed
+framework, dubbed as RM, consistently outperforms all other models with significantly large margins for all datasets. Note that RMna¨ıve
+indicates the basic version of distance rank merge described in Sec. 3.2 and that Old and New denote embedding models of gallery images.
+ImageNet-1K
+CIFAR-100
+Places-365
+Market-1501
+AUCmAP
+AUCCMC
+Gain
+AUCmAP
+AUCCMC
+Gain
+AUCmAP
+AUCCMC
+Gain
+AUCmAP
+AUCCMC
+Gain
+Old
+31.2
+49.7
+0%
+21.6
+34.3
+0%
+16.5
+30.7
+0%
+62.7
+82.7
+0%
+New
+51.3
+70.3
+100%
+47.4
+62.6
+100%
+23.4
+39.1
+100%
+77.3
+90.9
+100%
+RMna¨ıve (Ours)
+40.0
+63.9
+44%
+30.8
+49.1
+36%
+19.5
+35.8
+43%
+69.2
+87.0
+45%
+BCT [19]
+32.0
+46.3
+4%
+26.4
+43.5
+19%
+17.5
+37.0
+14%
+66.6
+84.3
+27%
+FCT [17]
+36.9
+58.7
+28%
+27.1
+49.4
+21%
+22.5
+37.3
+87%
+66.4
+84.2
+25%
+FCT (w/ side-info) [17]
+43.6
+65.0
+62%
+37.0
+53.9
+60%
+23.7
+38.3
+104%
+66.4
+84.4
+25%
+BiCT [20]
+35.1
+59.7
+19%
+29.0
+48.3
+29%
+19.0
+34.9
+36%
+65.0
+82.4
+16%
+RM (Ours)
+53.4
+68.1
+110%
+41.4
+60.7
+78%
+28.2
+41.7
+170%
+70.7
+87.6
+55%
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.30
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+mAP
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.50
+0.55
+0.60
+0.65
+0.70
+CMC (Top1 Acc.)
+(a) ImageNet-1K
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+mAP
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+CMC (Top1 Acc.)
+(b) CIFAR-100
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.175
+0.200
+0.225
+0.250
+0.275
+0.300
+mAP
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.32
+0.34
+0.36
+0.38
+0.40
+0.42
+CMC (Top1 Acc.)
+(c) Places-365
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.62
+0.64
+0.66
+0.68
+0.70
+0.72
+0.74
+0.76
+mAP
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.80
+0.82
+0.84
+0.86
+0.88
+0.90
+CMC (Top1 Acc.)
+(d) Market-1501
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.30
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+mAP
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.50
+0.55
+0.60
+0.65
+0.70
+Top1 Acc.
+Old
+New
+BCT
+FCT*
+FCT(w/ side-info)*
+BiCT
+RM_naïve (Ours)
+RM (Ours)
+Figure 7. mAP and CMC (Top-1 Acc.) results of our full framework in comparison to existing approaches. The numbers in the legend
+indicate either AUCmAP or AUCCMC scores.
+models. All transformation modules, ψ(·) and ρ(·), con-
+sist of 1 to 5 linear layer blocks, where each block is com-
+posed of a sequence of operations, (Linear → BatchNorm
+→ ReLU), except for the last block that only has a Lin-
+ear layer. Our algorithm does not use any side-information.
+Our modules are trained with the Adam optimizer [8] for 50
+epoch, where the learning rate is 1 × 10−4 at the beginning
+and decayed using cosine annealing [12]. Our frameworks
+are implemented with the Pytorch [16] library and we plan
+to release the source codes of our work.
+6.3. Results
+Homogeneous model upgrade
+We present the quantita-
+tive results in the homogeneous model upgrade scenario,
+where old and new models have the same architecture. We
+employ ResNet-50 for ImageNet and ResNet-18 for other
+datasets. Table 1 and Figure 7 compare the proposed frame-
+work, referred to as RM (Rank Merge), with existing com-
+patible learning approaches, including BCT [19], FCT [17],
+and BiCT [20]. As shown in the table, RM consistently out-
+performs all the existing compatible learning methods by
+remarkably significant margins in all datasets. BCT [19]
+learns backward compatible feature representations, which
+is backfill-free, but its performance gain is not impressive.
+
+FCT [17] achieves meaningful performance improvement
+by transforming old gallery features, but most of the gains
+come from side-information [2].
+For example, if side-
+information is not available, the performance gain of FCT
+drops from 62% to 28% on the ImageNet dataset. Also,
+such side-information is not useful for the re-identification
+dataset, Market-1501, mainly because the model for the
+side-information is trained for image classification using the
+ImageNet dataset, which shows its limited generalizability.
+On the other hand, although BiCT [20] takes advantage of
+online backfilling with less backfilling cost, it suffers from
+degraded final performance and negative flips in the mid-
+dle of backfilling. Note that RMna¨ıve, our na¨ıve rank merg-
+ing between old and new models, is already competitive to
+other approaches.
+Heterogeneous model upgrade
+We evaluate our frame-
+work in more challenging scenarios and present the results
+in Figure 8, where the old and new models have different
+architectures, e.g., ResNet-18 → ResNet-50 or ResNet-18
+→ ViT-B/32. In this figure, RMRQT (green line) denotes
+our ablative model trained with (9). Even in this setting,
+where both embedding spaces are more incompatible, our
+rank merge results from the old and new models still man-
+age to achieve a monotonous performance growth curve and
+RM improves the overall performance significantly further,
+which validates the robustness of our frameworks.
+Ablation study
+We analyze the results from the abla-
+tions of models for our cross-model contrastive learning.
+For compatible training, CL-S employs contrastive learn-
+ing within the backward system only as in (10) while our
+CMCL considers distance metrics from both backward and
+new retrieval systems simultaneously as in (11). For a more
+thorough ablation study, we also design and test another
+metric learning objective, called CL-M, which is given by
+LCL-M(xi, yi) = − log
+�
+yk=yi sold
+ik
+�
+yk=yi sold
+ik + �
+yk̸=yi sold
+ik
+− log
+�
+yk=yi snew
+ik
+�
+yk=yi snew
+ik + �
+yk̸=yi snew
+ik
+, (14)
+which conducts contrastive learning for both backward and
+new retrieval systems separately. Figure 9 visualizes the re-
+sults from the ablation studies, where CMCL consistently
+outperforms both CL-S and CL-M in various datasets and
+architectures. CL-M generally gives better merge results
+than CL-S because it calibrates the distances of new re-
+trieval system additionally.
+However, CL-M still suffers
+from negative flips because the distance metrics of both re-
+trieval systems are calibrated independently and not learned
+to be directly comparable to each other.
+On the other
+hand, CMCL improves overall performance curves con-
+sistently without negative flips.
+This validates that con-
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.3
+0.4
+0.5
+0.6
+mAP
+Old (0.223)
+New (0.513)
+RM (0.509)
+RM_RQT (0.372)
+RM_naïve (0.365)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+Top1 Acc.
+Old (0.436)
+New (0.703)
+RM (0.673)
+RM_RQT (0.631)
+RM_naïve (0.615)
+(a) ImageNet (ResNet-18 → ResNet-50)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+mAP
+Old (0.216)
+New (0.448)
+RM (0.420)
+RM_RQT (0.364)
+RM_naïve (0.309)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+Top1 Acc.
+Old (0.343)
+New (0.626)
+RM (0.611)
+RM_RQT (0.568)
+RM_naïve (0.514)
+(b) CIFAR-100 (ResNet-18 → ViT-B/32)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.65
+0.70
+0.75
+0.80
+mAP
+MCT (Ours) (0.747)
+Direct Alignment (0.709)
+Merge [Old+New] (0.722)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.82
+0.84
+0.86
+0.88
+0.90
+0.92
+Top1 Acc.
+MCT (Ours) (0.900)
+Direct Alignment (0.871)
+Merge [Old+New] (0.883)
+(c) Market-1501 (ResNet-18 → ResNet-50)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.175
+0.200
+0.225
+0.250
+0.275
+0.300
+0.325
+mAP
+Old (0.164)
+New (0.249)
+RM (0.292)
+RM_RQT (0.217)
+RM_naïve (0.208)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.32
+0.34
+0.36
+0.38
+0.40
+0.42
+Top1 Acc.
+Old (0.309)
+New (0.398)
+RM (0.427)
+RM_RQT (0.375)
+RM_naïve (0.366)
+(d) Places-365 (ResNet-18 → ResNet-50)
+Figure 8.
+Experimental results with heterogeneous model up-
+grades. Our na¨ıve rank merge between different architectures still
+achieves promising performance curves in various settings, and
+our full algorithm exhibits significantly better results.
+sidering the distance metrics of both systems simultane-
+ously helps to achieve better metric compatibility and con-
+sequently stronger merge results.
+7. Conclusion
+We presented a novel compatible training framework for
+effective and efficient online backfilling. We first addressed
+
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.45
+0.50
+0.55
+0.60
+mAP
+CMCL (Ours) (0.534)
+CL-M (0.487)
+CL-S (0.461)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.60
+0.62
+0.64
+0.66
+0.68
+0.70
+Top1 Acc.
+CMCL (Ours) (0.681)
+CL-M (0.648)
+CL-S (0.637)
+(a) ImageNet (ResNet-50 → ResNet-50)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.350
+0.375
+0.400
+0.425
+0.450
+0.475
+0.500
+mAP
+CMCL (Ours) (0.433)
+CL-M (0.411)
+CL-S (0.400)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.54
+0.56
+0.58
+0.60
+0.62
+Top1 Acc.
+CMCL (Ours) (0.617)
+CL-M (0.572)
+CL-S (0.594)
+(b) CIFAR-100 (ViT-B/32 → ViT-B/32)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.175
+0.200
+0.225
+0.250
+0.275
+0.300
+0.325
+mAP
+CMCL (Ours) (0.282)
+CL-M (0.228)
+CL-S (0.243)
+0
+20
+40
+60
+80
+100
+Backfill progress (%)
+0.30
+0.32
+0.34
+0.36
+0.38
+0.40
+0.42
+Top1 Acc.
+CMCL (Ours) (0.417)
+CL-M (0.383)
+CL-S (0.385)
+(c) Places-365 (ResNet-18 → ResNet-18)
+Figure 9. Ablation study of the cross-model contrastive learning
+loss on several datasets. CMCL outperforms other ablative mod-
+els, CL-M and CL-S, which validates that the distance calibration
+plays a crucial role for effective rank merging.
+the inherent trade-off between compatibility and discrim-
+inability, and proposed a practical alternative, online back-
+filling, to handle this dilemma. Our distance rank merge
+framework elegantly sidesteps this issue by bridging the gap
+between old and new models, and our metric-compatible
+learning further enhances the merge results with distance
+calibration. Our framework was validated via extensive ex-
+periments with significant improvement. We believe our
+work will provide a fundamental and practical foundation
+for promoting new directions in this line of research.
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+

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+Deep Learning for Mean Field Games with
+non-separable Hamiltonians
+Mouhcine Assoulia, Badr Missaouib
+aModeling, Simulation and Data Analysis Lab, Lot 660, Ben Guerir, 43150, Morocco
+bModeling,Simulation and Data Analysis Lab, Lot 660, Ben Guerir, 43150, Morocco
+Abstract
+This paper introduces a new method based on Deep Galerkin Methods (DGMs)
+for solving high-dimensional stochastic Mean Field Games (MFGs).
+We
+achieve this by using two neural networks to approximate the unknown so-
+lutions of the MFG system and forward-backward conditions. Our method
+is efficient, even with a small number of iterations, and is capable of han-
+dling up to 300 dimensions with a single layer, which makes it faster than
+other approaches.
+In contrast, methods based on Generative Adversarial
+Networks (GANs) cannot solve MFGs with non-separable Hamiltonians. We
+demonstrate the effectiveness of our approach by applying it to a traffic flow
+problem, which was previously solved using the Newton iteration method
+only in the deterministic case. We compare the results of our method to
+analytical solutions and previous approaches, showing its efficiency. We also
+prove the convergence of our neural network approximation with a single
+hidden layer using the universal approximation theorem.
+Keywords:
+Mean Field Games, Deep Learning, Deep Galerkin Method,
+Traffic Flow, Non-Separable Hamiltonian
+1. Introduction
+Mean Field Games (MFGs) are a widely studied topic that can model
+a variety of phenomena, including autonomous vehicles [1, 2], finance [3, 4],
+economics [5, 6, 7], industrial engineering [8, 9, 10], and data science [11, 12].
+MFGs are dynamic, symmetric games where the agents are indistinguishable
+but rational, meaning that their actions can affect the mean of the popu-
+lation. In the optimal case, the MFG system reaches a Nash equilibrium
+January 10, 2023
+arXiv:2301.02877v1  [cs.LG]  7 Jan 2023
+
+(NE), in which no agent can further improve their objective. MFGs are de-
+scribed by a system of coupled partial differential equations (PDEs) known
+as equation
+�
+�
+�
+−∂tφ − ν∆φ + H(x, ρ, ∇φ) = 0, in
+E1,
+∂tρ − ν∆ρ − div (ρ∇pH(x, ρ, ∇φ)) = 0, in
+E2,
+ρ(0, x) = ρ0(x),
+φ(T, x) = g(x, ρ(T, x)), in
+Ω,
+(1)
+where, E1 = (0, T] × Ω, E2 = [0, T) × Ω, Ω ⊂ Rd and g denotes the terminal
+cost. The Hamiltonian H with separable structure is defined as
+H(x, ρ, p) = infv{−p.v + L0(x, v)} − f0(x, ρ) = H0(x, p) − f0(x, ρ),
+(2)
+consisting of a forward-time Fokker-Planck equation (FP) and a backward-
+time Hamilton-Jacobi-Bellman equation (HJB), which describe the evolution
+of the population density (ρ) and the cost value (φ), respectively. The PDEs
+are defined in the domain E1 = (0, T]×Ω and E2 = [0, T). The Hamiltonian
+H has a separable structure and is defined as the infimum of the Lagrangian
+function L0, which is the Legendre transform of the Hamiltonian, minus the
+interaction function f0 between the population of agents. The MFG system
+also includes boundary conditions, with the initial density ρ(0, x) given by
+ρ0(x) and the terminal cost φ(T, x) given by g(x, ρ(T, x)). These boundary
+conditions apply in the domain Ω ⊂ Rd.
+One of the main challenges of MFGs is the viscosity problem, in addi-
+tion to the complexity of the PDEs and forward-backward conditions. Many
+methods for solving MFGs are limited to the deterministic setting (ν = 0).
+For example, the Newton iteration method has been applied to the prob-
+lem of traffic flow in [1], where a flexible machine learning framework was
+provided for the numerical solution of potential MFGs.
+While numerical
+methods do exist for solving the system of PDEs (1) [13, 14, 15, 16], they
+are not always effective due to computational complexity, especially in high
+dimensional problems. Deep learning methods, such as Generative Adver-
+sarial Networks (GANs) [17, 18], have been used to address this issue by
+reformulating MFGs as a primal-dual problem [19, 20, 14]. This approach
+uses the Hopf formula in density space [21] to establish a connection between
+MFGs and GANs. However, this method requires the Hamiltonian H to be
+separable in ρ and p. In cases where the Hamiltonian is non-separable, such
+as in traffic flow [1], it is not possible to reformulate MFGs as a primal-dual
+2
+
+problem. Recently, [22] proposed a policy iteration algorithm for MFGs with
+non-separable Hamiltonians using the contraction fixed point method.
+Contributions In this work, we present a new method based on DGM for
+solving stochastic MFG with a non-separable Hamiltonian. Inspired by the
+work [23, 24, 25], we approximate the unknown solutions of the system (1)
+by two neural networks trained simultaneously to satisfy each equation of the
+MFGs system and forward-backward conditions. While the GAN-based tech-
+niques are limited to problems with separable Hamiltonians, our algorithm,
+called New-Method, can solve any MFG system. Moreover, we prove the
+convergence of the neural network approximation with a single layer using a
+fundamental result of the universal approximation theorem. Then, we test
+the effectiveness of our New-Method through several numerical experiments,
+where we compare our results of New-Method with previous approaches to
+assess their reliability. At last, our approach is applied to solve the MFG
+system of traffic flow accounting for the stochastic case.
+Contents The structure of the rest of the paper is as follows: in Section 2,
+we introduce the main description of our approach. Section 3 examines the
+convergence of our neural network approximation with a single hidden layer.
+In Section 4, we present a review of prior methods. Section 5 investigates
+the numerical performance of our proposed algorithms.
+We evaluate our
+method using a simple analytical solution in Section 5.1 and compare it
+to the previous approach in Section 5.2. We also apply our method to the
+traffic flow problem in Section 5.3. Finally, we conclude the paper and discuss
+potential future work in Section 6.
+2. Methodology
+Our method involves using two neural networks, Nθ and Nω, to approx-
+imate the unknown variables ρ and φ, respectively. The weights for these
+networks are θ and ω. Each iteration of our method involves updating ρ
+and φ with the approximations from Nθ and Nω. To optimize the accuracy
+of these approximations, we use a loss function based on the residual of the
+first equation (HJB) to update the parameters of the neural networks. We
+repeat this process using the second equation (FP) and new parameters; see
+Figure 1. Both neural networks are simultaneously trained on the first equa-
+tion, and the results are then checked in the second equation, where they are
+3
+
+Figure 1: The learning mechanism of our method.
+fine-tuned until an equilibrium is reached. This equilibrium represents the
+convergence of the two neural networks and, therefore, the solution to both
+the Hamilton Jacobi Bellman equations and the Fokker-Planck equation.
+We have developed a solution for the problem of MFG systems 1 that
+does not rely on the Hamiltonian structure. Our approach involves using a
+combination of physics-informed deep learning [24] and deep hidden physics
+models [25] to train our model to solve high-dimensional PDEs that adhere to
+specified differential operators, initial conditions, and boundary conditions.
+Our model is also designed to adhere to general nonlinear partial differential
+equations that describe physical laws. To train our model, we define a loss
+function that minimizes the residual of the equation at randomly chosen
+points in time and space within the domain Ω.
+We initialize the neural networks as a solution to our system. We let:
+φω(t, x) = Nω(t, x),
+ρθ(t, x) = Nθ(t, x).
+(3)
+Our training strategy starts by solving (HJB). We compute the loss (4) at
+randomly sampled points {(tb1, xb1)}B1
+b1=1 from E1, and {xs1}S1
+s1=1 from Ω.
+Loss(HJB)
+total
+= Loss(HJB) + Loss(HJB)
+cond ,
+(4)
+4
+
+Update
+Input
+HJB
+On, Wn
+On+1,Wn+1
+On+2, Wn+2
+On+1, Wn+1
+FP
+Input
+Updatewhere
+Loss(HJB) = 1
+B1
+B1
+�
+b1=1
+���∂tφω(tb1, xb1) + ν∆φω(tb1, xb1)
+− H(xb1, ρθ(tb1, xb1), ∇φω(tb1, xb1))
+���
+2
+,
+and
+Loss(HJB)
+cond
+= 1
+S1
+S1
+�
+s1=1
+���φω(T, xs1) − g(xs1, ρθ(T, xs1))
+���
+2
+.
+We then update the weights of φω and ρθ by back-propagating the loss (4).
+We do the same to (FP) with the updated weights. We compute (5) at ran-
+domly sampled points {(tb2, xb2)}B2
+b2=1 from E2, and {xs2}S2
+s2=1 from Ω.
+Loss(FP)
+total = Loss(FP) + Loss(FP)
+cond ,
+(5)
+where
+Loss(FP) = 1
+B2
+B2
+�
+b2=1
+���∂tρθ(tb2, xb2) − ν∆ρθ(tb2, xb2)
+− div (ρθ(tb2, xb2)∇pH(xb2, ρθ(tb2, xb2), ∇φω(tb2, xb2)))
+���
+2
+,
+and
+Loss(FP)
+cond = 1
+S2
+S2
+�
+s2=1
+���ρθ(0, xs2) − ρ0(xs2)
+���
+2
+.
+Finally, we update the weights of φω and ρθ by back-propagating the loss (5);
+see Algorithm [1].
+3. Convergence
+Following the steps of [23], this section presents theoretical results that
+guarantee the existence of a single layer feedforward neural networks ρθ and
+φω which can universally approximate the solutions of (1). Denote
+L1(ρθ, φω) =
+���H1(ρθ, φω)
+���
+2
+L2(E1) +
+���φω(T, x) − φ(T, x)
+���
+2
+L2(Ω),
+(6)
+5
+
+Algorithm 1 New-Method
+Require: H Hamiltonian, ν diffusion parameter, g terminal cost.
+Require: Initialize neural networks Nω0 and Nθ0
+Train
+for n=0,1,2...,K-2 do
+Sample batch {(tb1, xb1)}B1
+b1=1 from E1, and {xs1}S1
+s1=1 from Ω
+L(HJB) ←
+1
+B1
+�B1
+b1=1
+���∂tφωn(tb1, xb1) + ν∆φωn(tb1, xb1)
+−H(xb1, ρθn(tb1, xb1), ∇φωn(tb1, xb1))
+���
+2
+.
+L(HJB)
+cond
+←
+1
+S1
+�S1
+s1=1
+���φωn(T, xs1) − g(xs1, ρθn(T, xs1))
+���
+2
+.
+Backpropagate Loss(HJB)
+total
+to ωn+1, θn+1 weights.
+Sample batch {(tb2, xb2)}B2
+b2=1 from E2, and {xs2}S2
+s2=1 from Ω.
+L(FP) ←
+1
+B2
+�B2
+b2=1
+���∂tρθn+1(tb2, xb2) ��� ν∆ρθn+1(tb2, xb2)
+− div(∇pH(xb2, ρθn+1(tb2, xb2), ∇φωn+1(tb2, xb2))
+×ρθn+1(tb2, xb2))
+���
+2
+.
+Lcond(FP) ←
+1
+S2
+�S2
+s2=1
+���ρθn+1(0, xs2) − ρ0(xs2)
+���
+2
+.
+Backpropagate Loss(FP)
+total to ωn+2 θn+2 weights.
+return θK, ωK
+where
+H1(ρθ, φω) = ∂tφω(t, x) + ν∆φω(t, x) − H(x, ρθ(x, t), ∇φω(t, x)).
+L2(ρθ, φω) =
+���H2(ρθ, φω)
+���
+2
+L2(E2) +
+���ρθ(0, x) − ρ0(x)
+���
+2
+L2(Ω),
+(7)
+and
+H2(ρθ, φω) = ∂tρθ(t, x)−ν∆ρθ(t, x)−div (ρθ(t, x)∇pH(x, ρθ(t, x), ∇φω(t, x))) .
+Denote ||f(x)||L2(E) =
+��
+E |f(x)|2dµ(x)
+� 1
+2 the norm on L2 and µ is a positive
+probability density on E. The aim of our approach is to identify a set of
+6
+
+parameters θ and ω such that the functions ρθ(x, t) and φω(x, t) minimizes
+the error L1(ρθ, φω) and L2(ρθ, φω). If L1(ρθ, φω) = 0 and L2(ρθ, φω) = 0,
+then ρθ(t, x) and φω(t, x) are solutions to (1). To prove the convergence of
+the neural networks, we use the results [26] on the universal approximation
+of functions and their derivatives. Define the class of neural networks with a
+single hidden layer and n hidden units,
+N n(σ) =
+�
+Φ(t, x) : R1+d �→ R : Φ(t, x) =
+n
+�
+i=1
+βiσ
+�
+α1,it +
+d
+�
+j=1
+αj,ixj + cj
+� �
+,
+Where
+θ = (β1, · · · , βn, α1,1, · · · , αd,n, c1, c1, · · · , cn) ∈ R2n+n(1+d),
+the vector of the parameter to be learned. The set of all functions imple-
+mented by such a network with a single hidden layer and n hidden units
+is
+N(σ) =
+�
+n≥1
+N n(σ),
+(8)
+We consider E a compact subset of Rd+1, from [26, Th 3]. we know that if
+σ ∈ C2 �
+Rd+1�
+is non constant and bounded, then N(σ) is uniformly 2-dense
+on E. This means by [26, Th 2] that for all u ∈ C1,2 �
+[0, T] × Rd�
+and ϵ > 0,
+there is fθ ∈ N(σ) such that:
+sup
+(t,x)∈E
+|∂tu(t, x) − ∂tfθ(t, x)| + max
+|a|≤2 sup
+(t,x)∈E
+��∂(a)
+x u(t, x) − ∂(a)
+x fθ(t, x)
+�� < ϵ. (9)
+To prove the convergence of our algorithm, we make the following assump-
+tions,
+• (H1): E1, E2 are compacts and consider the measures µ1, µ2, µ3, and µ4
+whose support is contained in E1, Ω, E2, and Ω respectively.
+• (H2): System (1) has a unique solution (φ, ρ) ∈ X × X such that:
+X =
+�
+u(t, x) ∈ C
+�
+¯
+[0, T] × Ω
+� �
+C1+η/2,2+η ([0, T] × Ω)
+with η ∈ (0, 1)and that
+sup
+(t,x)∈[0,T]×Ω
+2
+�
+k=1
+��∇(k)
+x u(t, x)
+�� < ∞
+�
+.
+7
+
+• (H3): H, ∇pH, ∇ppH, ∇ρpH are locally Lipschitz continuous in (ρ, p)
+with Lipschitz constant that can have at most polynomial growth in ρ
+and p, uniformly with respect to t, x.
+Remark 3.1. It is important to note that the nonlinear term of L2 can be
+simplified as follows,
+div(ρ∇pH(x, ρ, ∇φ)) = ∇pH(x, ρ, ∇φ)∇ρ + ∇pρH(x, ρ, ∇φ)∇ρ.ρ
++
+�
+i,j
+∇pipjH(x, ρ, ∇φ)(∂xjxiφ)ρ.
+Theorem 3.1. Let consider N(σ) where σ is C2 �
+Rd+1�
+, non constant and
+bounded.
+Suppose (H1), (H2), (H3) hold.
+Then for every ϵ1, ϵ2 > 0,
+there exists two positives constant C1, C2 > 0 and there exists two functions
+(ρθ, φω) ∈ N(σ) × N(σ), such that,
+Li(ρθ, φω) ≤ Ci(ϵ1 + ϵ2),
+for
+i = {1, 2}.
+The proof of this theorem is in Appendix A.
+Now we have L1(ρn
+θ, φn
+ω) �→ 0, and L2(ρn
+θ, φn
+ω) �→ 0 as n �→ ∞ but it does
+not necessarily imply that (ρn
+θ, φn
+ω) �→ (ρ, ω) is the unique solution.
+We now prove, under stronger conditions, the convergence of the neural net-
+work, (ρn
+θ, φn
+w) to the solution (ρ, φ) of the system 1 as n → ∞.
+To avoid some difficulties, we add homogeneous boundary conditions that
+assume the solution is vanishing on the boundary. The MFG system (1)
+writes
+�
+�
+�
+�
+�
+�
+�
+−∂tφ − ν div (a1(∇φ)) + γ(ρ, ∇φ) = 0, in
+ΩT,
+∂tρ − ν div (a2(∇ρ)) − div (a3(ρ, ∇φ)) = 0, in
+ΩT,
+ρ(0, x) = ρ0(x),
+φ(T, x) = g(x, ρ(T, x)), in
+Ω,
+ρ(t, x) = φ(t, x) = 0, in
+Γ,
+(10)
+where, ΩT = (0, T) × Ω, Γ = (0, T) × ∂Ω and
+a1(t, x, ∇φ) = ∇φ,
+a2(t, x, ∇ρ) = ∇ρ,
+a3(t, x, ρ, ∇φ) = ρ∇pH(x, ρ, ∇φ),
+γ(t, x, ρ, ∇φ) = H(x, ρ, ∇φ),
+8
+
+a1 : ΩT × RN → RN, a2 : ΩT × RN × RN → RN, a3 : ΩT × R × RN → RN
+and γ : ΩT × R × RN → R are Caratheodory functions.
+Then we introduce the approximate problem of the system (10) as
+�
+�
+�
+�
+�
+�
+�
+−∂tφn
+ω − ν div (a1(∇φn
+ω)) + γ(ρn
+θ, ∇φn
+ω) = 0, in
+ΩT,
+∂tρn
+θ − ν div (a2(∇ρn
+θ)) − div (a3(ρn
+θ, ∇φn
+ω) = 0, in
+ΩT,
+ρn
+θ(0, x) = ρ0(x),
+φn
+ω(T, x) = g(x, ρn
+θ(T, x)), in
+Ω,
+ρn
+θ(t, x) = φn
+ω(t, x) = 0. in
+Γ,
+(11)
+Let us first introduce some definitions.
+Let r ≥ 1. In the sequel we denote by Lr �
+0, T; W 1,r
+0 (Ω)
+�
+the set of functions
+u such that u ∈ Lr (ΩT), u(t, ·) ∈ W 1,r
+0 (Ω). The space Lr �
+0, T; W 1,r
+0 (Ω)
+�
+is
+equipped with the norm
+∥u∥Lr(0,T;W 1,r
+0
+(Ω)) :=
+�� T
+0
+�
+Ω
+|∇u(x, t)|rdxdt
+� 1
+r
+,
+is a Banach space. For s, r ≥ 1, the space V s,r
+0
+(ΩT) := L∞ (0, T; Ls(Ω)) ∩
+Lr �
+0, T; W 1,r
+0 (Ω)
+�
+endowed with the norm
+∥ϕ∥V s,r
+0
+(ΩT ) := ess sup
+0≤t≤T
+∥ϕ(., t)∥Ls(Ω) + ∥ϕ∥Lr(0,T;W 1,r
+0
+(Ω)),
+is also a Banach space.
+For this convergence, we make the following set of assumptions,
+• (H4): There is a constant µ > 0 and positive functions κ(t, x), λ(t, x)
+such that for all (t, x) ∈ ΩT, we have
+∥a3(t, x, ρ, p)∥ ≤ µ(κ(t, x) + ∥p∥), and |γ(t, x, ρ, p)| ≤ λ(t, x)∥p∥,
+with κ ∈ L2 (ΩT) , λ ∈ Ld+2 (ΩT) .
+• (H5): a3(t, x, ρ, p) and γ(t, x, ρ, p) are Lipschitz continuous in (t, x, ρ, p) ∈
+ΩT×R×Rd uniformly on compacts of the form
+�
+(t, x) ∈ ¯ΩT, |ρ| ≤ C, |p| ≤ C
+�
+.
+• (H6): There is a positive constant α > 0 such that
+a3(t, x, ρ, p)p ≥ α|p|2.
+9
+
+• (H7): For every n ∈ N, ρn
+θ, φn
+ω ∈ C1,2 �¯ΩT
+�
+. In addition, (ρn
+θ)n∈N , (φn
+ω)n∈N ∈
+L2 (ΩT) .
+Theorem 3.2. Under previous assumptions (H4)-(H7), if we assume that
+(10) has a unique bounded solution (φ, ρ) ∈ V 2,2
+0
+×V 2,2
+0
+, then (φn
+ω, ρn
+θ) converge
+to (φ, ρ) strongly in Lp (ΩT) × Lp (ΩT) for every p < 2.
+The proof of this theorem is in Appendix B. Related Works
+4. Related Works
+GANs: Generative adversarial networks, or GANs, are a class of ma-
+chine learning introduced in 2014 [27] that have been successful in generat-
+ing images and processing data [28, 29, 30]. In recent years, there has been
+increasing interest in using GANs for financial modeling as well [31]. GANs
+consist of two neural networks, a generator network, and a discriminator
+network, that work against each other in order to generate samples from a
+specific distribution. As described in various sources [27, 32, 33], the goal is
+to reach equilibrium for the following problem,
+min
+G max
+D
+�
+Ex∼Pdata(x)[log(D(x)] + Ez∼Pg(z)[log(1 − D(G(z))]
+�
+,
+(12)
+where Pdata(x) is the original data and Pg(z) is the noise data. In (12), the
+goal is to minimize the generator’s output (G) and maximize the discrimina-
+tor’s output (D). This is achieved by comparing the probability of the original
+data Pdata(x) being correctly identified by the discriminator D with the prob-
+ability of the generated data G produced by the generator using noise data
+Pg(z) being incorrectly identified as real by the discriminator 1 − D(G(z)).
+Essentially, the discriminator is trying to accurately distinguish between real
+and fake data, while the generator is attempting to create fake data that can
+deceive the discriminator.
+APAC-Net: In [17], the authors present a method (APAC-Net) based
+on GANs for solving high-dimensional MFGs in the stochastic case. They use
+of the Hopf formula in density space to reformulate the MFGs as a saddle-
+point problem given by,
+inf
+ρ(x,t) sup
+φ(x,t)
+�
+Ez∼P(z),t∼Unif[0,T][∂tφ(ρ(t, z), t) + ν∆φ(ρ(t, z), t)
+− H(ρ(t, z), ∇φ)] + Ez∼P(z)φ(0, ρ(0, z)) − Ex∼ρT φ(T, x)
+�
+,
+(13)
+10
+
+where
+H(x, p) = infv{−p.v + L(x, v)}.
+In this case, we have a connection between the GANs and MFGs, since
+(13) allows them to reach the Kantorovich-Rubenstein dual formulation of
+Wasserstein GANs [33] given by,
+min
+G max
+D {Ex∼Pdata(x)[(D(x)] − Ez∼Pg(z)[(D(G(z))]},
+s.t. ||∇D|| ≤ 1.
+(14)
+Finally, we can use an algorithm similar to GANs to solve the problems of
+MFGs. Unfortunately, we notice that the Hamiltonian in this situation has a
+separable structure. Due to this, we cannot solve the MFG-LWR system (to
+be detailed in section 5.3). In general, we cannot solve the MFGs problems,
+where its Hamiltonian is non-separable, since we cannot reformulate MFGs
+as 13.
+MFGANs: In [18, 17], the connection between GANs and MFGs is
+demonstrated by the fact that equation (13) allows them to both reach the
+Kantorovich-Rubinstein dual formulation of Wasserstein GANs, as described
+in reference [33]. This is shown in equation (12), which can be solved using
+an algorithm similar to those used for GANs. However, it is not possible to
+solve MFGs problems with non-separable Hamiltonians, as they cannot be
+reformulated as in equation (13). This is because the Hamiltonian in these
+cases has a separable structure, which prevents the solution of the MFG-
+LWR system (to be discussed in section 5.3).
+DGM-MFG: In [34], section 4 discusses the adaptation of the DGM al-
+gorithm to solve mean field games, referred to as DGM-MFG. This method
+is highly versatile and can effectively solve a wide range of partial differential
+equations due to its lack of reliance on the specific structure of the problem.
+Our own work is similar to DGM-MFG in that we also utilize neural net-
+works to approximate unknown functions and adjust parameters to minimize
+a loss function based on the PDE residual, as seen in [34] and [18]. However,
+our approach, referred to as New-Method, differs in the way it is trained.
+Instead of using the sum of PDE residuals as the loss function and SGD for
+optimization, we define a separate loss function for each equation and use
+ADAM for training, following the approach in [18]. This modification allows
+11
+
+for faster and more accurate convergence.
+Policy iteration Method: To the best of our knowledge, [22] was the
+first to successfully solve systems of mean field game partial differential equa-
+tions with non-separable Hamiltonians. They proposed two algorithms based
+on policy iteration, which involve iteratively updating the population distri-
+bution, value function, and control. These algorithms only require the solu-
+tion of two decoupled, linear PDEs at each iteration due to the fixed control.
+This approach reduces the complexity of the equations, but it is limited to
+low-dimensional problems due to the computationally intensive nature of the
+method. In contrast, our method utilizes neural networks to solve the HJB
+and FP equations at each iteration, allowing for updates to the population
+distribution and value function in each equation without the limitations of
+[22].
+5. Numerical Experiments
+To evaluate the effectiveness of the proposed algorithm [1], we use the
+example provided in [17], as it has an explicitly defined solution structure
+that allows for easy numerical comparison. We compare the performance of
+New-Method, APAC-Net’s MFGAN, and DGM-MFG on the same data to
+assess their reliability. Additionally, we apply New-Method to the traffic flow
+problem [19], which is characterized by its non-separable Hamiltonian [20],
+to determine its ability to solve this type of problem in a stochastic case.
+5.1. Analytic Comparison
+We test our method by comparing it to a simple example of the analytic
+solution used to test the effectiveness of Apac-Net [17].
+For the sake of
+simplicity, we take the spatial domain Ω = [−2, 2]d, the final time T = 1,
+and without congestion (γ = 0). For
+H0(x, p) = ||p||2
+2
+− β ||x||2
+2 ,
+f0(x, ρ) = γln(ρ),
+g(x) = α ||x||2
+2
+− (νdα + γ d
+2ln α
+2πν),
+(15)
+and ν = β = 1, where
+α = −γ +
+�
+γ2 + 4ν2β
+2ν
+= 1.
+12
+
+The corresponding MFG system is:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−∂tφ − ∆φ + ||∇φ||2
+2
+− ||x||2
+2
+= 0,
+∂tρ − ∆ρ − div (ρ∇φ) = 0,
+ρ(0, x) = ( 1
+2π)
+d
+2e− ||x||2
+2 ,
+φ(T, x) = x2
+2 − d,
+(16)
+and the explicit formula is given by
+φ(t, x) = ||x||2
+2
+− d.t,
+ρ(t, x) = ( 1
+2π)
+d
+2e− ||x||2
+2 .
+(17)
+Test 1: We consider the system of PDEs [16] in one dimension (d =
+1).
+To obtain results, we run Algorithm [1] for 5.103 iterations, using a
+minibatch of 50 samples at each iteration. The neural networks employed
+have three hidden layers with 100 neurons each, and utilize the Softplus
+activation function for Nω and the Tanh activation function for Nθ. Both
+networks use ADAM with a learning rate of 10−4 and a weight decay of 10−3.
+We employ ResNet as the architecture of the neural networks, with a skip
+connection weight of 0.5. The numerical results are shown in Figure 2, which
+compares the approximate solutions obtained by New-Method to the exact
+solutions at different time states.
+To evaluate the performance of New-Method, we compute the relative error
+between the model predictions and the exact solutions on a 100 × 100 grid
+within the domain [0, 1] × [−2, 2]. Additionally, we plot the HJB and FP
+residual loss, as defined in Algorithm [1], to monitor the convergence of our
+method (see Figure 3).
+Test 2: In this experiment, we use a single hidden layer with vary-
+ing numbers of hidden units (nU) for both neural networks. As previously
+shown in section 2, the number of hidden units can affect the convergence of
+the model. To verify this, we repeat the previous test using the same hyper-
+parameters and a single hidden layer but with different numbers of hidden
+units. The relative error between the model predictions and the exact solu-
+tions is then calculated on a 100×100 grid within the domain [0, 1]×[−2, 2],
+as shown in Figure 4.
+Test 3: We solve the MFG system [16] for dimensions 2, 50, and 100.
+Figure 5 shows the residuals of the HJB and FP equations over 5.104 iter-
+ations. A minibatch of 1024, 512, and 128 samples were used for d=100,
+d=50, and d=2, respectively. The neural networks had three hidden layers
+13
+
+Figure 2: The exact solution and prediction calculated by New-Method in dimension one
+at t=(0.25, 0.5, 0.75 ).
+Figure 3: The relative error for ρ, φ for the figure on the left. On the right, the HJB, FP
+Loss.
+14
+
+t=0.25
+175
+pexact
+150
+p new method
+125
+ exact
+@ new method
+1D0
+0.75
+0.50
+0.25
+0.00 
+0.25
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1D
+15
+2Dt=0.5
+150
+pexact
+125
+p new method
+LDO -
+ exact
+@ new method
+0.75
+0.50 
+0.25
+0.0
+0.25
+0.50
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1D
+15
+2Dt=0.75
+125
+pexact
+1D0
+p new method
+0.75
+ exact
+@ new method
+0.50 
+0.25
+0.0
+0.25
+0.50
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1D
+15
+2DThe relative errorof p and 
+10
+Relative errar
+0
+2400
+ODE
+50D0
+iberationsResiduals for Fp and HjB cguation
+loss FP
+10° 
+loss HJB
+10-1
+Losg
+10-
+10
+10
+0
+2400
+ODE
+400
+50D0
+IeratiorsFigure 4: The relative error for ρ , φ in 1-dimension for nU=(2, 5, 10, 20, 50).
+with 100 neurons each and utilized the Softplus activation function for Nω
+and the Tanh activation function for Nθ. Both networks used ADAM with
+a learning rate of 10−4, weight decay of 10−3, and employed ResNet as their
+architecture with a skip connection weight of 0.5. The results were obtained
+by recording the residuals every 100 iterations and using a rolling average
+over 5 points to smooth out the curves.
+Figure 5: The loss HJB and FP equation for d=(2,10,100)
+Test 4: In this test, we use the same setup as before, but with a single
+layer of 100 neurons instead of multiple layers. We keep all other neural
+network hyperparameters unchanged.
+This test is meant to demonstrate
+that a single layer can perform better than multiple layers, even when the
+dimension increases, as seen in section 2. Figure 6 shows improved results
+compared to the previous test, even with few iterations, which allows for
+faster computation times.
+15
+
+Residuals for Fp equation
+10-2 1
+.- d=2
+- d=50
+E-OT
+. d=100
+10+
+lenprs
+10
+ 10-
+107 
+10-8 
+10-
+0
+14000
+24000
+ODIDE
+4D00
+50000
+iberationa cf AdamThe relative error phi
+100
+nU=2
+10-1
+nU=5
+nU=10
+nU=20
+nU=50
+0
+2400
+30
+400
+50D0
+iberatiors of AdamlThe relative eror rha
+nU=2
+nU=5
+= nU=10
+100
+nU=20
+nU=50
+XYYY
+101
+0
+2400
+30D0
+400
+50D0
+iberatior of AdamlResiduals for HjB eguation
+- d=2
+103
+-- d=50
+... d=100
+107 
+10
+100
+10-1
+10~2
+0
+1ADO0
+24000
+ODIDE
+4D00
+50000
+iberationa cf AdamFigure 6: The loss HJB and FP equation with a minibatch of 128, 512, and 1024 samples
+for d=2, d=50, and d=(100,200,300), respectively.
+5.2. Comparison
+In previous sections, we introduced and discussed four methods for solv-
+ing MFGs: APAC-Net, MFGAN, DGM-MFG, and New-Method. Here, we
+compare these approaches to assess their performance. For APAC-Net, it is
+only possible to compare the cost values φ due to the unavailability of the
+density function. In APAC-Net, the generator neural network represents ρ,
+which generates the distribution. In order to compare the results, we need
+to use kernel density estimation to transform the distribution into a density,
+which is only an estimate. We use the simple example from the analytic so-
+lution with d = 1 and T = 1 for this comparison. The two neural networks in
+this comparison have three hidden layers with 100 neurons each, and utilize
+ResNet as their architecture with a skip connection weight of 0.5. They also
+use the Softplus activation function for Nω and the Tanh activation function
+for Nθ. For training APAC-Net, MFGAN, and New-Method, we use ADAM
+with a learning rate of 10−4 and a weight decay of 10−3 for both networks.
+For training DGM-MFG, we use SGD initialized with a value of 10−3 and a
+weight decay of 10−3 for both networks.
+We run the four algorithms for 5.103 iterations, using a minibatch of 50 sam-
+ples at each iteration. The relative error between the model predictions and
+the exact solutions is then calculated on a 100 × 100 grid within the domain
+[0, 1] × [−2, 2], as shown in Figure 7.
+5.3. Application (Traffic Flow):
+In a study published in [1], the authors focused on the longitudinal speed
+control of autonomous vehicles. They developed a mathematical model called
+16
+
+Residual for HjB eguation
+... d=2
+104
+- d=50
+- d=100
+- d=200
+102
+— d=300
+Bso1 lenpgad
+100
+102 ,
+0
+2400
+14000
+keratiorsResidual for Fp eguation
+101
+10-2
+10-3
+10-4
+d=2
+101
+d=50
+d=100
+10-0
+d=200
+d=300
+0
+2400
+410
+6+DO
+14000
+keratiorsFigure 7: comparison between APAC-Net, MFGAN, DGM-MFG, and New-Method.
+a Mean Field Game (MFG) to solve a traffic flow problem for autonomous
+vehicles and demonstrated that the traditional Lighthill-Whitham-Richards
+(LWR) model can be used as a solution to the MFG-LWR model described
+by the following system of equations:
+MFG − LWR
+�
+�
+�
+�
+�
+�
+�
+Vt + U(ρ)Vx − 1
+2V 2
+x = 0,
+ρt + (ρu)x = 0,
+u = U(ρ) − Vx,
+VT = g(·, ρT),
+ρ(·, 0) = ρ0.
+(18)
+Here, ρ, V , and u represent the density, optimal cost, and speed function,
+respectively, and the Greenshields density-speed relation is given by U(ρ) =
+umax(1 − ρ/ρjam), where ρjam is the jam density and umax is the maximum
+speed. By setting ρjam = 1 and umax = 1, the authors generalized the MFG-
+LWR model to include a viscosity term µ > 0, resulting in the following
+system:
+MFG − LWR
+�
+�
+�
+Vt + ν∆V − H(x, p, ρ) = 0,
+ρt − ν∆ρ − div(∇pH(x, p, ρ)ρ) = 0,
+VT = g(·, ρT),
+ρ(·, 0) = ρ0.
+(19)
+In this model, ρ and V represent the density and optimal cost function,
+respectively, and H is the Hamiltonian with a non-separable structure given
+by
+H(x, p, ρ) = 1
+2||p||2 − (1 − ρ)p,
+with p = Vx,
+(20)
+where p = Vx.
+The authors solved the system in (19) using the New-
+ton Iteration method for the deterministic case (ν = 0) with a numerical
+17
+
+Comparison of p
+- New-Method
+100
+MFGAN
+DGM-MFG
+Relative errar
+10-1
+0
+2400
+ODE
+50D0
+iberationsComparison of 
+..- New-Method
+MFGAN
+DGM-MFG
+100
+APAC net
+Relative errar
+10-1
+0
+24D0
+ODE
+5400
+iberationsmethod that considers only a finite number of discretization points to re-
+duce computational complexity.
+In this work, we propose a new method
+using a neural network to approximate the unknown and solve the prob-
+lem in the stochastic case, while also avoiding the computational complexity
+of the previous method. To evaluate the performance of the new method,
+we consider the traffic flow problem defined by the MFG-LWR model in 19
+with a non-separable Hamiltonian in (20) on the spatial domain Ω = [0, 1]
+with dimension d = 1 and final time T = 1.
+The terminal cost g is
+set to zero and the initial density ρ0 is given by a Gaussian distribution,
+ρ0(x) = 0.2 − 0.6 exp
+�
+−1
+2
+� x−0.5
+0.1
+�2�
+. The aim is to investigate the perfor-
+mance of the new method, called the ”New-Method,” in solving this traffic
+flow problem.
+The corresponding MFG system is,
+�
+�
+�
+�
+�
+�
+�
+Vt + ν∆V − 1
+2||Vx||2 + (1 − ρ)Vx = 0
+ρt − ν∆ρ − div((Vx − (1 − ρ))ρ) = 0
+ρ(x, 0) = 0.2 − 0.6 exp( −1
+2 ( x−0.5
+0.1 )2),
+φ(x, T) = 0.
+(21)
+We study the deterministic case (ν = 0) and stochastic case (ν = 0.5). We
+represent the unknown solutions by two neural networks Nω and Nθ, which
+have a single hidden layer of 50 neurons. We use the ResNet architecture
+with a skip connection weight of 0.5. We employ ADAM with learning rate
+4 × 10−4 for Nω and 5 × 10−4 for Nθ and weight decay of 10−4 for both
+networks, batch size 100, in both cases ν = 0 and ν = 0.5 we use the
+activation function Softmax and Relu for Nω and Nθ respectively. In Figure
+8 we plot over different times the density function, the optimal cost, and the
+speed which is calculated according to the density and the optimal cost [1]
+by the following formula,
+u = umax(1 − ρ/ρjam) − Vx
+where, we take the jam density ρjam = 1 and the maximum speed umax = 1
+and 104 iterations.
+++ In Figure (9), we plot the HJB, FP residual loss for ν = 0 and ν = 0.5,
+which helps us monitor the convergence of our method. Unfortunately, we
+do not have the exact solution to compute the error. To validate the results
+of Figure (8), we use the fundamental traffic flow diagram, an essential tool
+to comprehend classic traffic flow models. Precisely, this is a graphic that
+18
+
+t=0
+t=0.5
+t=1
+Figure 8: The solution of the problem MFG-LWR by New-Method for (ν = 0) and (ν =
+0.5) at t=(0,0.5,1).
+Figure 9: The loss HJB and FP equation for (ν = 0) and (ν = 0.5).
+19
+
+v=o
+10-1 ,
+.- loss FP
+loss HJB
+10-3
+i
+10-
+10~7
+10-5
+1022,
+1013
+0
+2400
+400
+14000
+keratiorsv=05
+101
+-
+loss FP
+loss HJB
+102
+10-3
+Bso1 lenpgad
+10
+100
+10~7
+10-8
+0
+2400
+6+DO
+14000
+keratiorsdensity
+speed
+Optimal costdensity
+speed
+Optimal costdensity
+speed
+Optimal costdensity
+speed
+Optimal costdensity
+speed
+Optimal costdensity
+speed
+Optimal cost0.8
+0.7
+0.6 
+0.5 
+density
+0.4 
+speed
+EO
+Optimal cost
+0.2 
+0.1 
+0.0 
+0'0
+0.2
+0.4
+0.6 
+0.8
+10V=0.5
+0.8
+0.7
+0.6
+0.5
+density
+0.4
+speed
+Optimalcost
+0.3
+0.2
+0.1
+0.0
+00
+0.2
+0.4
+0.6
+0.8
+10
+xV=O
+0.8
+0.7
+0.6
+0.5
+density
+0.4 :
+speed
+EO
+Optimal cost
+0.2
+0.1
+0.0
+00
+0.2
+0.4
+0.6
+0.8
+1.0
+x0.8
+0.7
+0.6
+0.5
+density
+0.4 :
+speed
+EO
+Optimalcost
+0.2
+0.1
+0.0
+0.0
+0.2 
+0.4 
+0.6
+8'0
+1.0
+x0.8
+0.7
+0.6
+0.5
+density
+0.4
+speed
+Optimalcost
+EO
+0.2
+0.1
+0.0
+00
+0.2 
+0.4
+0.6
+0.8
+1.00.8
+0.7
+0.6 
+0.5 
+density
+0.4
+speed
+E0
+Optimal cost
+0.2
+0.1 
+0.0 
+0'0
+0.2
+0.4
+9:0
+0.8
+10displays a link between road traffic flux (vehicles/hour) and the traffic density
+(vehicles/km) [35, 36, 37]. We can find this diagram numerically [1] such as
+its function q is given by,
+q(t, x) = ρ(t, x)u(t, x).
+Figure (10) shows the fundamental diagram of our results.
+t=0
+t=0.5
+t=1
+Figure 10: Fundamental diagram for ν = (0, 0.5) at t = (0, 0.5, 1).
+6. Conclusion
+• We present a new method based on the deep galerkin method (DGM)
+for solving high-dimensional stochastic mean field games (MFGs). The
+key idea of our algorithm is to approximate the unknown solutions by
+two neural networks that were simultaneously trained to satisfy each
+equation of the MFGs system and forward-backward conditions.
+• Consequently, our method shows better results even in a small number
+of iterations because of its learning mechanism. Moreover, it shows the
+20
+
+densitydenstbydensitydensitydansitydensityV=0.5
+0.24
+0.22
+flow
+0.20
+0.18
+0.16
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+density0.24
+0.22
+MOU
+0.20
+0.18
+0.16
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+0.55
+density0.168
+0.167
+0.166
+10
+0.165
+0.164
+0.163
+0.162
+0.204
+0.206
+0.208
+0.210
+0.212
+0.214
+densityV=O
+0.24
+0.22
+0.18
+0.16
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+density0.24
+0.22
+0.20
+0.18
+0.16
+0.20
+0.25
+0.30
+SEO
+0.40 
+density0.180
+0.175
+0.170
+0.165
+0.160
+0.20
+0.21
+0.22
+0.23
+0.24
+densitypotential of up to 300 dimensions with a single layer, which gives more
+speed to our method.
+• we proved that as the number of hidden units increases, the neural
+networks converge to the MFG solution.
+• Comparison with the previous methods shows the efficiency of our ap-
+proach even with multilayer neural networks.
+• Test on traffic flow problem in the deterministic case gives results sim-
+ilar to the newton iteration method, showing that it can solve this
+problem in the stochastic case.
+To address the issue of high dimensions in the problem, we used a neural
+network but found that it took a significant amount of time.
+While our
+approach has helped to reduce the time required, it is still not fast enough.
+Therefore, we are seeking an alternative to neural networks in future research
+to improve efficiency.
+Appendix A. Proof of Theorem 3.1.
+Denote N(σ) the space of all functions implemented by such a network
+with a single hidden layer and n hidden units, where σ in C2 �
+Rd+1�
+non-
+constant, and bonded. By (H1) we have that for all ρ, φ ∈ C1,2 �
+[0, T] × Rd�
+and ε1, ε2 > 0, There is ρθ, φω ∈ N(σ) such That,
+sup
+(t,x)∈E1
+|∂tφ(t, x) − ∂tφω(t, x)|
++ max
+|a|≤2
+sup
+(t,x)∈E1
+��∂(a)
+x φ(t, x) − ∂(a)
+x φω(t, x)
+�� < ϵ1
+(A.1)
+sup
+(t,x)∈E2
+|∂tρ(t, x) − ∂tρθ(t, x)|
++ max
+|a|≤2
+sup
+(t,x)∈E2
+��∂(a)
+x ρ(t, x) − ∂(a)
+x ρθ(t, x)
+�� < ϵ2
+(A.2)
+From (H3) we have that (ρ, p) �→ H(x, ρ, p) is locally Lipschitz continuous
+in (ρ, p), with Lipschitz constant that can have at most polynomial growth
+in ρ and p, uniformly with respect to t, x. This means that
+|H(x, ρ, p) − H(x, γ, s)| ≤
+�
+|ρ|q1/2 + |p|q2/2 + |γ|q3/2 + |s|q4/2�
+× (|ρ − γ| + |p − s|).
+21
+
+with some constants 0 ≤ q1, q2, q3, q4 < ∞. As a result, we get using H¨older
+inequality with exponents r1, r2,
+�
+E1
+|H (x, ρθ, ∇xφω) − H (x, ρ, ∇φ)|2 dµ1(t, x)
+≤
+�
+E1
+(|ρθ(t, x)|q1 + |∇φω(t, x)|q2 + |ρ(t, x)|q3 + |∇φ(t, x)|q4)
+×
+�
+|ρθ(t, x) − ρ(t, x)|2 + |∇φω(t, x) − ∇φ(t, x)|2�
+dµ1(t, x)
+≤
+� �
+E1
+(|ρθ(t, x)|q1 + |∇φω(t, x)|q2 + |ρ(t, x)|q3 + |∇φ(t, x)|q4)r1dµ1(t, x)
+�1/r1
+×
+� �
+E1
+(|ρθ(t, x) − ρ(t, x)|2 + |∇φω(t, x) − ∇φ(t, x)|2)r2dµ1(t, x)
+�1/r2
+≤ C1
+� �
+E1
+(|ρθ(t, x) − ρ(t, x)|q1 + |∇φω(t, x) − ∇φ(t, x)|q2
++ |ρ(t, x)|q1∨q3 + |∇φ(t, x)|q2∨q4)r1dµ1(t, x)
+�1/r1
+×
+� �
+E1
+(|ρθ(t, x) − ρ(t, x)|2 + |∇φω(t, x) − ∇φ(t, x)|2)r2dµ1(t, x)
+�1/r2
+≤ C1
+�
+ϵq1
+1 + ϵq2
+2 + sup
+E1
+|ρ|q1∨q3 + sup
+E1
+|∇φ|q2∨q4
+�
+(ϵ2
+1 + ϵ2
+2)
+≤ C1(ϵ2
+1 + ϵ2
+2),
+where the constant C1 < ∞ may change from line to line and qi ∨ qj =
+max{qi, qj}. In the two last steps we used A.1, A.2 and (H2). We recall
+that,
+H1(ρθ, φω) = ∂tφω(t, x) + ν∆φω(t, x) − H(x, ρθ(t, x), ∇φω(t, x)).
+22
+
+Note that H1(ρ, φ) = 0 for ρ, θ that solves the system of PDEs,
+L1(ρθ, φω) =
+���H1(ρθ, φω)
+���
+2
+L2(E1) +
+���φω(T, x) − φ(T, x)
+���
+2
+L2(Ω)
+=
+���H1(ρθ, φω) − H1(ρ, φ)
+���
+2
+L2(E1) +
+���φω(x, T) − g(x, ρθ(x, T))
+���
+2
+L2(Ω)
+≤
+�
+E1
+|∂tφω(t, x) − ∂tφ(t, x)|2 dµ1(t, x)
++ |ν|
+�
+E1
+|∆φω(t, x) − ∆φ(t, x)|2 dµ1(t, x)
++
+�
+E1
+|H (x, ρθ, ∇φω) − H (x, ρ, ∇φ)|2 dµ1(t, x)
++
+�
+Ω
+|φω(T, x) − φ(T, x)|2dµ2(t, x)
+≤C1(ϵ2
+1 + ϵ2
+2)
+for an appropriate constant C1 < ∞. In the last step, we use A.1, A.2 and
+the previous result.
+For L2 we use remark 3.1 to simplified the nonlinear term,
+div(ρ∇pH(x, ρ, ∇φ)) = α1(x, ρ, ∇φ) + α2(x, ρ, ∇φ) + α3(x, ρ, ∇φ),
+where,
+α1(x, ρ, ∇φ) = ∇pH(x, ρ, ∇φ)∇ρ,
+α2(x, ρ, ∇φ) = ∇pρH(x, ρ, ∇φ)∇ρ.ρ,
+α3(x, ρ, ∇φ) =
+�
+i,j
+∇pipjH(x, ρ, ∇φ)(∂xjxiφ)ρ.
+In addition, from (H3) we have also ∇pH(x, ρ, p), ∇pρH(x, ρ, p), and ∇ppH(x, ρ, p)
+are locally Lipschitz continuous in (ρ, p). Then, we have after an application
+of Holder inequality, for some constant C2 < ∞ that may change from line
+23
+
+to line,
+�
+E2
+|α1 (x, ρθ, ∇φω) − α1(x, ρ, ∇φ)|2 dµ3(t, x)
+=
+�
+E2
+|∇pωH (x, ρθ, ∇φω) ∇ρθ − ∇pH(x, ρ, ∇φ)∇ρ|2 dµ3(t, x)
+≤
+�
+E2
+���
+�
+∇pωH (x, ρθ, ∇φω) − ∇pH(x, ρ, ∇φ)
+�
+∇ρ
+���
+2
+dµ3(t, x)
++
+�
+E2
+���∇pωH (x, ρθ, ∇φω) (∇ρθ − ∇ρ)
+���
+2
+dµ3(t, x)
+≤ C2
+��
+E2
+���∇pωH (x, ρθ, ∇φω) − ∇pH(x, ρ, ∇φ)
+���
+2r1dµ3 (t, x)
+�1/r1
+×
+� �
+E2
+|∇ρ|2r2dµ3(t, x)
+�1/r2 + C2
+��
+E2
+���∇pωH (x, ρθ, φω)
+���
+2s1dµ3(t, x)
+�1/s1
+×
+��
+E2
+|∇ρθ − ∇ρ|2s2 dµ3(t, x)
+�1/s2
+≤ C2
+� �
+E2
+|∇ρ|2r2dµ3(t, x)
+�1/r2
+×
+� �
+E2
+(|ρθ(t, x) − ρ(t, x)|q1 + |∇φω(t, x) − ∇φ(t, x)|q2
++ |ρ(t, x)|q1∨q3 + |∇φ(t, x)|q2∨q4)v1r1dµ3(t, x)
+�1/v1r1
+×
+� �
+E2
+(|ρθ(t, x) − ρ(t, x)|2 + |∇xφω(t, x) − ∇xφ(t, x)|2)v2r2dµ3(t, x)
+�1/v2r2
++ C2
+��
+E2
+���∇pωH (x, ρθ, φω)
+���
+2s1dµ3(t, x)
+�1/s1
+×
+��
+E2
+|∇ρθ − ∇ρ|2s2 dµ3(t, x)
+�1/s2
+≤ C2(ϵ2
+1 + ϵ2
+2)
+where in the last steps, we followed the computations previously. We do
+same for α2(x, ρ, ∇φ) and α3(x, ρ, ∇φ), we obtain for a C2 < ∞,
+�
+E2
+��� div(ρθ∇pωH(x, ρθ, ∇φω)) − div(ρ∇pH(x, ρ, ∇φ))
+���
+2
+dµ3(t, x)
+≤ C2(ϵ2
+1 + ϵ2
+2).
+24
+
+We recall that,
+H2(ρθ, φω) = ∂tρθ(t, x)−ν∆ρθ(t, x)−div (ρθ(t, x)∇pH(x, ρθ(t, x), ∇φω(t, x)))
+Note that H2(ρ, φ) = 0 for ρ, θ that solves the system of PDEs, then we have,
+L2(ρθ, φω) =
+���H2(ρθ, φω)
+���
+2
+L2(E2) +
+���ρθ(0, x) − ρ0(x)
+���
+2
+L2(Ω)
+=
+���H2(ρθ, φω) − H2(ρ, φ)
+���
+2
+L2(E2) +
+���ρθ(0, x) − ρ0(x)
+���
+2
+L2(Ω)
+≤
+�
+E2
+|∂tρθ(t, x) − ∂tρ(t, x)|2 dµ3(t, x)
++ |ν|
+�
+E2
+|∆ρθ(t, x) − ∆ρ(t, x)|2 dµ3(t, x)
++
+�
+E2
+��� div(ρθ∇pωH(x, ρθ, ∇φω)) − div(ρ∇pH(x, ρ, ∇φ))
+���
+2
+dµ3(t, x)
++
+�
+Ω
+|ρθ(0, x) − ρ0(x)|2dµ4(t, x)
+≤C2(ϵ2
+1 + ϵ2
+2)
+for an appropriate constant C2 < ∞. The proof of theorem 3.1 is complete
+after rescaling ϵ1 and ϵ2
+Appendix B. Proof of Theorem 3.2.
+We follow the method used in [23] for a single PDE. (See also section 4
+in [38] for a coupled system). Let us denote the solution of problem 11 by.
+�
+ˆρn
+θ, ˆφn
+ω
+�
+∈ V = V 2,2
+0
+× V 2,2
+0
+. Due to Conditions (H4) − (H6) and by using
+lemma 1.4 [39] on each equation then, there exist, C1, C2 such that:
+∥ˆρn
+θ∥V 2,2
+0
+≤ C1
+∥ˆφn
+ω∥V 2,2
+0
+≤ C2
+These applies and gives that the both sequence {ˆρn
+θ}n∈N, {ˆφn
+ω}n∈N are uni-
+formly bounded with respect to n in at least V . These uniform energy bounds
+25
+
+imply the existence of two subsequences, (still denoted in the same way)
+{ˆρn
+θ}n∈N, {ˆφn
+ω}n∈N and two functions ρ, φ in L2 �
+0, T; W 1,2
+0 (Ω)
+�
+such that,
+ˆρn
+θ → ρ weakly in L2 �
+0, T : W 1,2
+0 (Ω)
+�
+ˆφn
+ω → φ weakly in L2 �
+0, T : W 1,2
+0 (Ω)
+�
+Next let us set q = 1 +
+d
+d+4 ∈ (1, 2) and note that for conjugates, r1, r2 > 1
+such that 1/r1 + 1/r2 = 1
+�
+ΩT
+���γ
+�
+t, x, ˆρn
+θ, ∇ˆφn
+ω
+����
+q
+≤
+�
+ΩT
+|λ|q ���∇ˆφn
+ω
+���
+q
+≤
+��
+ΩT
+|λ|r1q
+�1/r1 ��
+ΩT
+���∇ˆφn
+ω
+���
+r2q�1/r2
+Let us choose r2 = 2/q > 1. Then we calculate r1 =
+r2
+r2−1 =
+2
+2−q. Hence,
+we have that r1q = d + 2. Recalling the assumption λ ∈ Ld+2 (ΩT) and the
+uniform bound on the ∇ˆφn
+ω we subsequently obtain that for q = 1 +
+d
+d+4,
+there is a constant C < ∞ such that
+�
+ΩT
+���γ
+�
+t, x, ˆρn
+θ, ∇ˆφn
+ω
+����
+q
+≤ C
+On the other hand, it is obvious that a1 is bounded uniformly then, according
+to the HJB equation of 11, we have
+�
+∂t ˆφn
+ω
+�
+n∈N is bounded uniformly with
+respect to n in L2 (0, T; W −1,2(Ω)). Then we can extract a subsequence, (still
+denoted in the same way)
+�
+∂t ˆφn
+ω
+�
+n∈N such that
+∂t ˆφn
+θ → ∂tφ weakly in L2 �
+0, T; W −1,2(Ω)
+�
+Also, it will be shown that
+∂tˆρn
+θ → ∂tρ weakly in L2 �
+0, T; W −1,2(Ω)
+�
+Since the problem is nonlinear, the weak convergence of ˆφn
+ω and ˆρn
+θ in the
+space L2 �
+0, T; W 1,2
+0 (Ω)
+�
+is not enough in order to prove that φ and ρ are a
+solution of problem 10. To do this, we need the almost everywhere conver-
+gence of the gradients for a subsequence of the approximating solutions ˆφn
+ω
+and ˆρn
+θ.
+26
+
+However, the uniform boundedness of {ˆφn
+ω}n∈N and {ˆρn
+θ}n∈N in L2 �
+0, T; W 1,2
+0 (Ω)
+�
+and their weak convergence to φ and ρ respectively in that space, allows us
+to conclude, by using Theorem 3.3 of [40] on each equation, that
+∇ˆφn
+ω → ∇φ almost everywhere in ΩT.
+∇ˆρn
+θ → ∇ρ almost everywhere in ΩT.
+Hence, we obtain that {ˆφn
+ω}n∈N and {ˆρn
+θ}n∈N converges respectively to φ and
+ρ strongly in Lp �
+0, T; W 1,p
+0 (Ω)
+�
+for every p < 2. It remains to discuss the
+convergence of φn
+ω − ˆφn
+ω and ρn
+θ − ˆρn
+θ to zero. By last step of proof theorem 7.3
+[23] we get
+�
+φn
+ω − ˆφn
+ω
+�
+n∈N and {ρn
+θ − ˆρn
+θ}n∈N goes to zero strongly in Lp (ΩT)
+for every p < 2. Finally we conclude the proof of the convergence in Lp (ΩT)
+for every p < 2
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+

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+arXiv:2301.13647v1  [physics.data-an]  31 Jan 2023
+Bayesian estimation of information-theoretic metrics for sparsely sampled distributions
+Angelo Piga,∗ Lluc Font-Pomarol,† Marta Sales-Pardo,‡ and Roger Guimer`a§
+(Dated: February 1, 2023)
+Estimating the Shannon entropy of a discrete distribution from which we have only observed a small sample is
+challenging. Estimating other information-theoretic metrics, such as the Kullback-Leibler divergence between
+two sparsely sampled discrete distributions, is even harder. Existing approaches to address these problems
+have shortcomings: they are biased, heuristic, work only for some distributions, and/or cannot be applied to all
+information-theoretic metrics. Here, we propose a fast, semi-analytical estimator for sparsely sampled distribu-
+tions that is efficient, precise, and general. Its derivation is grounded in probabilistic considerations and uses a
+hierarchical Bayesian approach to extract as much information as possible from the few observations available.
+Our approach provides estimates of the Shannon entropy with precision at least comparable to the state of the
+art, and most often better. It can also be used to obtain accurate estimates of any other information-theoretic
+metric, including the notoriously challenging Kullback-Leibler divergence. Here, again, our approach performs
+consistently better than existing estimators.
+I.
+INTRODUCTION
+Information theory is gaining momentum as a methodolog-
+ical framework to study complex systems. In network sci-
+ence, information theory provides rigorous tools to predict
+unobserved links [1] and to infer community structure [2].
+In neuroscience, Shannon entropy of spike train distributions
+characterizes brain activity from neural responses [3], while
+mutual information identifies correlations between brain stim-
+uli and responses [4]. Recently, the Kullback-Leibler diver-
+gence [5] and its regularized version, the Jensen-Shannon dis-
+tance, have also been successfully used in a wide variety of
+contexts: in cognitive science as a measure of “surprise,” to
+quantify and predict how human attention is oriented between
+changing screen images [6]; in quantitative social science,
+in combination with topic models, to track the propagation
+of political and social discourses [7, 8] or to understand the
+emergence of social disruption from the analysis of judicial
+decisions [9]; and in machine learning, at the intersection be-
+tween the statistical physics of diffusive processes, probabilis-
+tic models and deep neural networks [10].
+Information theoretical metrics are measured on distribu-
+tions. In practice, a distribution ρ over the possible states
+of a system, as well as functions F(ρ) of this distribution
+(such as Shannon entropy or other metrics), have to be in-
+ferred from experimental observations. However, this infer-
+ence process is difficult for many real complex systems since,
+due to experimental limitations, the observations are often
+sparse, and statistical estimates of the distribution ρ and its
+functions can be severely biased. Here, we focus on the par-
+ticular yet important case of discrete (or categorical) distribu-
+tions ρi, i = 1, . . . , K, where K is the number of possible
+∗ [email protected]; Department of Chemical Engineering, Universi-
+tat Rovira i Virgili, Tarragona 43007, Catalonia.
+† [email protected];
+Department of Chemical Engineering,
+Universitat
+Rovira i Virgili, Tarragona 43007, Catalonia.
+‡ [email protected]; Department of Chemical Engineering, Universitat
+Rovira i Virgili, Tarragona 43007, Catalonia.
+§ [email protected]; Department of Chemical Engineering, Universitat
+Rovira i Virgili, Tarragona 43007, Catalonia.; ICREA, Barcelona 08010,
+Catalonia.
+states (or categories), which is known and fixed. Inferences
+about ρ and any function must be based on ni, the number
+of observations in the i-th state (with N = �
+i ni the sam-
+ple size) and, in the undersampled regime we are interested
+in, N ≲ K. The challenge is thus, from the sparse observa-
+tions {ni}, to infer the probability ρi of each category i and
+estimate metrics F(ρ).
+A theoretically well-founded approach to tackle this prob-
+lem is provided by the principles of conditional probability,
+encapsulated in Bayes’ theorem [11]. This framework is in
+general preferable because of its transparency—it requires
+that all assumptions of the underlying generative model for the
+data are made explicit, expressed via the choice of a likelihood
+function and a prior distribution that reflects the knowledge
+about the system before observing any data. In probabilistic
+reasoning, the combination of observations and prior distri-
+bution provides an updated (posterior) probability distribution
+of the quantity under study. Other estimation strategies make
+implicit assumptions and often provide only point estimates,
+as opposed to full distributions.
+A class of expressive generative models for categorical dis-
+tributions amenable to a Bayesian framework is the well-
+studied family of Dirichlet distributions. However, as Ne-
+menmann, Shafee, and Bialek (henceforth NSB) pointed out
+in [12], when sample sizes are small (N ≲ K), the inferred
+Shannon entropy is tightly determined by the specific parame-
+ters one chooses for the Dirichlet model; therefore, inaccurate
+choices result in severe biases of the Shannon entropy esti-
+mates. To overcome this problem, they introduced a mixture
+of Dirichlet models, which results in a very precise estimator
+of the Shannon entropy that works for a wide variety of distri-
+butions, even in the sparse sampling regime N ≲ K [12, 13].
+Although, in terms of precision, NSB can be considered the
+state of the art for estimating the Shannon entropy, it does not
+provide estimates for the distribution ρ. For this reason, its ap-
+plicability is limited to estimating the Shannon entropy (and
+related information theoretic quantities like mutual informa-
+tion and Jensen-Shannon distance, which can be expressed in
+terms of entropies). By contrast, it cannot be used to estimate
+the Kullback-Leibler divergence. To cover this gap, Hausser
+and Strimmer derived a James-Stein-type shrinkage estimator
+for ρ [14] (henceforth HS), which has the advantage of being
+
+2
+analytical and applicable to any information-theoretic metric,
+but at the price of making implicit ad hoc assumptions, and
+of being less precise than NSB for the Shannon entropy, and
+lacking error estimation.
+Here, we propose an alternative fast, semi-analytical es-
+timator for distributions that is efficient, precise, and gen-
+eral. Its derivation is grounded in probabilistic considerations,
+without any ad hoc assumptions. We consider Dirichlet gen-
+erative models and use a hierarchical Bayesian approach to
+extract as much information as possible from the few observa-
+tions at hand. In the case of Shannon entropy, we can estimate
+the expected value and higher order moments with precision
+at least comparable to the NSB estimator, and most often bet-
+ter. Additionally, because our method provides estimates of
+the probability distribution, it can be used to obtain accurate
+estimations of the Kullback-Leibler divergence. In this case
+our approach also performs equally or better than existing es-
+timators.
+II.
+BACKGROUND
+Let us consider a system with K possible output states
+whose observations follow an unknown discrete distribu-
+tion ρ = {ρi; i = 1, . . . , K} with �
+i ρi = 1.
+The vector
+n = {ni; i = 1, . . . , K} represents the number of times each
+state was observed in a set of �
+i ni = N independent obser-
+vations of the system. We also consider a function F(ρ) of ρ,
+such as, for example, the Shannon entropy
+S(ρ) = −
+K
+�
+i=1
+ρi log ρi ,
+(1)
+which we want to estimate from the set of observations.
+The posterior distribution over the values of the function F
+given the observed counts n is
+p (F|n) =
+�
+dρ δ (F − F(ρ)) p(ρ|n) ,
+(2)
+where p(ρ|n) is the posterior of the distribution ρ given the
+counts n. We further assume that the prior over distributions
+depends on a parameter β, which becomes a hyperparameter
+of our generative model. Then, using the laws of conditional
+probability, we can write the posterior p(ρ|n, β) as
+p(ρ|n, β) = p(n|ρ, β) p(ρ|β)
+p(n|β)
+,
+(3)
+where p(n|ρ, β) is the likelihood, p(ρ|β) is the prior over
+distributions, and p(n|β) =
+�
+dρ p(n|ρ) p(ρ|β) is the evi-
+dence and acts as normalization factor. The likelihood is the
+probability of the empirical observations n given ρ; for in-
+dependent multinomial samples, the probability of observing
+an event of type i is ρi, and the full likelihood is the prod-
+uct p(n|ρ, β) = p(n|ρ) = N! �K
+i ρni
+i /ni! and, given ρ, it
+is independent of the hyperparameter β. The prior p(ρ|β)
+expresses the probability of each distribution ρ prior to ob-
+serving any data, and plays a crucial role in the discussion be-
+low. Symmetric Dirichlet distributions are convenient priors
+because they are a generative model for a broad class of dis-
+crete distributions. Additionally, they have been widely used
+in this setting [15], and are parametrized as follows
+p(ρ|β) =
+1
+BK(β)
+K
+�
+i=1
+ρβ−1
+i
+,
+BK(β) = Γ(β)K
+Γ(βK) ,
+(4)
+where Γ is the gamma function, while the hyperparameter β is
+a real, positive number known as the concentration parameter.
+In the first row of Fig. 1, examples of categorical distributions
+sampled from symmetric Dirichlet priors are shown.
+Besides being very expressive, Dirichlet priors are conju-
+gate distributions of categorical likelihoods, meaning that the
+posterior is still a Dirichlet distribution, a property that often
+makes the inference via Eqs. (3) and (2) analytically tractable.
+For example, when F(ρ) = ρ, Dirichlet priors lead to ex-
+pected posterior probabilities ⟨ρi⟩ given by the widely-used
+generalized Laplace’s formula
+⟨ρi⟩ =
+ni + β
+N + Kβ .
+(5)
+It is worth noting the improvement of Eq. (5) with respect to
+the maximum likelihood (or frequency) estimator ρi = ni/N,
+which is recovered by the former in the limit β → 0. In par-
+ticular, Laplace’s formula assigns non zero probability to non
+observed states, a desirable property whose advantage will be-
+come evident later, when estimating Kullback-Leibler diver-
+gences. This example also illustrates how non-Bayesian ap-
+proaches to inference make implicit and non-trivial assump-
+tions, in this case assuming β → 0 amounts to assuming that
+infinitely concentrated distributions ρ are a priori much more
+plausible than more homogeneous ones.
+Going back to the estimation of F from the observations
+n, and given Eq. (5), one may be tempted to directly plug
+the value of ⟨ρi⟩ in the explicit expression of F(ρ) to get a
+point estimate. However, this is just an approximation; the
+exact procedure consists in finding and using the whole pos-
+terior p(F|n). Specifically, the expected value of this pos-
+terior ⟨F⟩ =
+�
+dF F p(F|n) minimizes the mean-squared
+error [16], and its mode is a consistent estimator, meaning
+that it converges to the true value of F(ρ) when the num-
+ber of observations increases, regardless of the prior and, in
+particular, regardless of the hyperparameter β. Wolpert and
+Wolf in Refs. [16, 17] provided analytical formulas for all
+the moments of p(F|n) when F is the Shannon entropy and
+for Dirichlet priors (we report the formula for the mean in
+Eq. (15) and for the second moment in Appendix B).
+However, an unbiased estimation of F is not guaranteed for
+small samples. This is often the case for Dirichlet priors, es-
+pecially when the parameter β is unknown. Several options
+for the value of β have been proposed in literature, each one
+suitable to some specific case but deficient in others (for a dis-
+cussion, refer to Refs. [12, 14]). In [12], NSB suggested that,
+when samples are scarce, any attempt to find a single universal
+β is hopeless; the fundamental reason being that categorical
+distributions generated by a Dirichlet have a Shannon entropy
+that is narrowly determined by, and monotonically dependent
+
+3
+on, β. In other words, for small samples, the posterior distri-
+bution (2) is dominated by the prior. To overcome this prob-
+lem, Refs. [12, 13] proposed, as the prior pNSB(ρ), an infinite
+mixture of Dirichlet priors
+pNSB(ρ) ∝
+�
+dβ pNSB(β) p(ρ|β) ,
+(6)
+where the weights pNSB(β) were set so as to obtain a flat prior
+over entropies S, and have the functional form
+pNSB(β) ∝ d E[S|ni = 0, β]
+dβ
+= Kψ1(Kβ +1)−ψ1(β +1) ,
+(7)
+where E[S|n, β] is the expected entropy given the observa-
+tions n, and then E[S|ni = 0, β] is the expected entropy of the
+distributions ρ generated from a symmetric Dirichlet priors
+(that is if there are no observations), with fixed β and K, and
+ψm(x) =
+� d
+dx
+�m+1 log Γ(x) are the polygamma functions.
+The NSB prior leads to very accurate estimates of the Shannon
+entropy, and can be considered the state of the art. Even if best
+suited for situations in which the number of states K is known
+and fixed, it is quite versatile and has been later extended for
+countable infinite number of states [18] and further optimized
+for binary states [19] and long tail distributions [18]. Other es-
+timators, for example, the Chao-Shen estimator [20], perform
+at most as well as the NSB (or its derivatives), but never better
+(see [14] for a comprehensive review). Additionally, given an
+estimator of S, a number of other quantities can be indirectly
+estimated. For example, the mutual information M between
+two distributions ρ and σ is M(ρ ; σ) = S(ρ)+S(σ)−S(π),
+where π is the joint distribution of ρ and σ [21]. Similar re-
+lations can be derived for Jensen-Shannon distance and other
+information-theoretic quantities [8] [22].
+However, consider the estimation of the Kullback-Leibler
+divergence (DKL) between two distributions ρ and σ with the
+same dimension K
+DKL(ρ∥σ) =
+K
+�
+i=1
+ρi log2
+ρi
+σi
+.
+(8)
+To estimate DKL from samples n = {ni; i = 1, . . . , K} from
+ρ, and m = {mi; i = 1, . . . , K} from σ, one cannot use the
+NSB approach. First, DKL is not a combination of the Shan-
+non entropies of the two underlying distributions ρ and σ.
+Second, DKL is unbounded, and any attempt to find a hyper-
+prior in the spirit of Eq. (7) results in improper hyperpriors.
+Finally, with the NSB prior one renounces to any estimation
+of β and, in turn, to a good a point estimation of DKL by
+means of Laplace’s formula.
+III.
+HIERARCHICAL BAYES POINT ESTIMATE FOR β
+Here, we address these limitations of the NSB estimator
+while maintaining and even improving its performance. We
+posit that the success of the NSB approach stems, not from
+mixing infinitely many values of the concentration parameter
+β, but rather from the flexibility to accommodate for any par-
+ticular value of β. Indeed, we surmise that, in general, only
+a narrow interval of β values are compatible with a given ob-
+servation n and therefore contribute to the mixture, whereas
+most others do not contribute. Motivated by this, we propose
+an approach that aims to directly estimate the value of β that
+most contributes to the posterior given the data n.
+First, we observe that the posterior p(ρ|n) can be written as
+p(ρ|n) =
+�
+dβ p(ρ|n, β) p(β|n)
+=
+�
+dβ p(n|ρ) p(ρ|β)
+p(n|β)
+p(β|n) ,
+(9)
+where we have applied Bayes’ rule, and the fact that n condi-
+tioned on ρ is independent of β, so that p(n|ρ, β) = p(n|ρ).
+Then, we assume that the conditional distribution p(β|n) is
+very peaked around a given value β⋆ , so that the posterior
+p(ρ|n) can be approximated as
+p(ρ|n) ≈ p(n|ρ) p(ρ|β⋆)
+p(n|β⋆)
+.
+(10)
+This approximation, sometimes referred to as empirical
+Bayes, is a point estimate for the fully hierarchical probabilis-
+tic model given by p(n|ρ) and p(ρ|β). Eq. (10) is identical to
+Eq. (3), with the difference that the concentration parameter
+is now the most likely value of β given the observed counts n,
+that is,
+β⋆ = argmax
+β
+p(β|n) = argmax
+β
+p(n|β) p(β)
+p(n)
+,
+(11)
+where p(n|β) =
+�
+dρ p(n|β, ρ)p(ρ|β). For Dirichlet priors
+(Eq. (4)), β∗ satisfies (see Appendix A)
+K
+�
+i=1
+ni−1
+�
+m=0
+1
+m + β⋆ −
+N−1
+�
+m=0
+K
+m + Kβ⋆ +
+1
+p(β⋆)
+d p(β)
+d β
+���
+β⋆ = 0 ,
+(12)
+which is the key analytical result of this paper.
+The hyperprior p(β) reflects our prior knowledge about the
+shape of the distribution of the hyperparameter. To be com-
+pletely agnostic in this regard, we can use a uniform hyper-
+prior
+pU(β) =
+1
+∆β = const. ,
+∆β = βmax − βmin ,
+(13)
+with cut-offs 0 < βmin < βmax < ∞. In this case, the deriva-
+tive term in Eq. (12) disappears. The NSB hyperprior (7) is a
+valid alternative; in this case, the last term in Eq. (12) is (see
+appendix A for details)
+1
+pNSB(β∗)
+d pNSB(β)
+d β
+����
+β∗ = K2ψ2(kβ⋆ + 1) − ψ2(β⋆ + 1)
+Kψ1(kβ⋆ + 1) − ψ1(β⋆ + 1) .
+(14)
+Despite the complex appearance of Eq. (12), β∗ is not
+hard to obtain numerically, giving a computational improve-
+ment with respect the NSB estimator, whose algorithm is
+involved and has higher computational costs [23].
+The
+source code of the implementations in Python is available at
+https://github.com/angelopiga/info-metric-estimation/.
+
+4
+1
+200
+400
+600
+800
+1000
+i
+0.1
+0.2
+0.3
+ρ
+i
+Dirichlet
+:
+ β
+=
+0.01, S
+=
+0.394
+1
+200
+400
+600
+800
+1000
+i
+0.0025
+0.0050
+0.0075
+Dirichlet
+:
+ β
+=
+1, S
+=
+0.936
+1
+200
+400
+600
+800
+1000
+i
+0.001
+0.002
+Dirichlet
+:
+ β
+=
+10, S
+=
+0.993
+1
+200
+400
+600
+800
+1000
+i
+0.005
+0.010
+ρ
+i
+half empty Dirichlet
+:
+ β
+=
+1, S
+=
+0.838
+1
+200
+400
+600
+800
+1000
+i
+10
+−4
+10
+−3
+10
+−2
+10
+−1
+Zipf
+:
+ a
+=
+1.001, S
+=
+0.751
+1
+200
+400
+600
+800
+1000
+i
+0.0025
+0.0050
+0.0075
+Bimodal
+:
+ S
+=
+0.854
+FIG. 1. Examples of target distributions. First row: three categorical distributions sampled from uniform Dirichlet with β = 0.01, 1, 10,
+respectively. Second row: a categorical distribution sampled from a uniform Dirichlet, β = 1, but where half bins are set to zero; Zipf’s dis-
+tribution with exponent a = 1.001; binomial distribution: two gaussians with {mean, standard deviation} respectively {10, 20} and {100, 5},
+are concatenated and then discretized over a histogram of 1000 categories.
+IV.
+RESULTS
+We test our method in a variety of scenarios and com-
+pare the results with the main alternative available estima-
+tors, the NSB [12, 13] and the Hausser-Strimmer (HS) [14].
+In our experiments, we generate synthetic target distributions
+and sample multinomial counts {ni} from those distributions.
+We fix K = 1000 and generate samples of increasing size
+N = 20, . . ., 10000. After calculating β⋆ from (12), we es-
+timate the Shannon entropy S and the Kullback-Leibler di-
+vergence DKL. For each case, we repeat this procedure 1000
+times; we always report averages over these repetitions [24].
+As target distributions (see Fig. 1 as reference) we consider
+categorical distributions that are both typical in the Dirich-
+let prior (that is, they are generated by a symmetric Dirich-
+let prior; we use several values of concentration parameter
+β = 0.01, 1, 10) and atypical in the Dirichlet prior (that is,
+they cannot be attributed to or have a negligible probability of
+being generated from a symmetric Dirichlet prior). Among
+the latter, we consider: (i) distributions with added struc-
+tural zeroes (that is, we sample from a symmetric Dirich-
+let prior with a given β, but half of the categories are then
+forced to have zero probability) [25]; (ii) Bimodal distribu-
+tions, which represent, for example, the degree distributions
+of core-periphery complex networks [26]; (iii) Zipf’s distri-
+bution, ubiquitous in nature, in biological as well as social
+systems [27], characterized by probabilities ρi ∝ i−a, with a
+exponent a ≥ 1 [28].
+A.
+Shannon entropy
+To estimate the posterior p(S|n) of the Shannon entropy we
+use the exact formulas of its moments, derived in Refs. [16,
+17] (later refined in Ref. [18]). The first moment is given by
+E[S|n, β] =
+�
+dρ S(ρ|β) p(ρ|n)
+= ψ0(N + Kβ + 1)
+−
+K
+�
+i=1
+ni + β
+N + Kβ ψ0(ni + β + 1) .
+(15)
+In Appendix B we also show the expression of the standard
+deviation.
+In practice, given a dataset n we calculate the most prob-
+able β⋆ from Eq. (12) by assuming either a flat hyperprior,
+Eq. (13), or the NSB hyperprior, Eq. (7). Then, we compute
+the required moments of the Shannon entropy; we indicate
+the estimated values of the Shannon entropy as S(β⋆
+flat) and
+S(β⋆
+NSB), respectively. In Figs. 2 and 3, we show that our
+estimator with a flat hyperprior is the most accurate estima-
+tor overall. In particular, S(β⋆
+flat) is consistently more accu-
+rate than the NSB estimator, except in the deep sparse regime
+N < 30 of two of the distributions atypical in the Dirichlet
+prior, where it is comparable but slightly less accurate. The
+Bayesian estimators also behave better than the HS estimator
+SHS except for very uniform distributions sampled from the
+Dirichlet prior with β = 10. Overall, the S(β⋆
+flat) has little
+bias often even in the very sparse regime and for distributions
+atypical in the Dirichlet prior. It is also interesting to note
+that both S(β⋆
+flat) and S(β⋆
+NSB) have a more regular scaling
+behavior, in the convergence toward the true values as N in-
+creases, in particular when compared with NSB and HS for
+Zipf’s distribution.
+We also analyze the variability of the Shannon entropy es-
+timates, as measured by the root mean squared error (insets
+in Figs. 2 and 3). This analysis reveals that, besides having
+less bias, the S(β⋆
+flat) estimator has a variability that is typi-
+cally comparable to or smaller than the other estimators. It is
+
+5
+10
+2
+10
+3
+10
+4
+N
+0.0
+0.2
+0.4
+ΔS
+rel
+Dirichlet : K
+=
+1000,  β
+=
+0.01, S
+true
+=
+0.422
+S
+true
+β
+⋆
+flat
+β
+⋆
+NSB
+NSB
+HS
+10
+2
+10
+3
+10
+4
+N
+10
+−2
+10
+−1
+RMSE
+10
+2
+10
+3
+10
+4
+N
+−0.15
+−0.10
+−0.05
+0.00
+0.05
+ΔS
+rel
+Dirichlet : K
+=
+1000,  β
+=
+1, S
+true
+=
+0.939
+10
+2
+10
+3
+10
+4
+N
+10
+−2
+10
+−1
+RMSE
+10
+2
+10
+3
+10
+4
+N
+−0.20
+−0.15
+−0.10
+−0.05
+0.00
+ΔS
+rel
+Dirichlet : K
+=
+1000,  β
+=
+10, S
+true
+=
+0.993
+10
+2
+10
+3
+10
+4
+N
+10
+−3
+10
+−2
+10
+−1
+RMSE
+FIG. 2.
+Shannon entropy estimation for distributions typical in
+a Dirichlet prior, for β
+=
+0.01, 1, 10 and sample size N
+=
+25, . . . , 10000.
+Each point corresponds to an average over 1000
+samples. The Strue in the titles serves as a reference and indicates
+the average over the entropies of the runs. Main plots: relative er-
+rors of entropies ∆Srel = (Sest − Strue)/Strue. Insets: roots
+mean-squared errors (note the logarithmic scale in both axes). Black
+squares: our estimator with β⋆ from a flat hyperprior. Cyan pluses:
+our estimator but with β⋆ from NSB hyperprior. Pink upper triangle:
+NSB estimator. Red crosses: Hausser-Strimmer plug-in estimator.
+Here and in the rest of figures, the standard-errors bars of the main
+plots are smaller then symbols and are not shown.
+also worth noting that, differently from Bayesian estimators,
+for which all the moments can be estimated also from a single
+sample, the HS estimator is limited to a point estimate of the
+mean value of Shannon entropy.
+Note that, contrary to what one may expect, SNSB differs
+from our estimate S(β⋆
+NSB) in that the latter is always smaller
+for small samples.
+This happens because the NSB hyper-
+prior (7) is a positive monotonically-decreasing function that
+assigns higher probabilities to smaller β’s, while the Shannon
+entropy of distributions sampled from a symmetric Dirichlet
+is a monotonically-increasing function of β. However, it is
+not the same estimating β⋆ with the NSB hyperprior and then
+plug it in (15) or directly estimating the Shannon entropy with
+the NSB prior (6) and the latter in fact provides better results.
+On the other side, S(β⋆
+flat) and SNSB should not substantially
+differ, being based on the same first principles of estimation.
+The differences are attributable to the numerical and compu-
+tational difficulties in implementing the NSB approach that
+required both a fine discretization over β and solving as many
+equations (15) as β’s, which have to be finally integrated with
+weights given by the hyperprior (7), in contrast with our ap-
+proach, which needs solving just Eq. (12) and Eq. (15) once.
+10
+2
+10
+3
+10
+4
+N
+−0.1
+0.0
+0.1
+ΔS
+rel
+Half empty Dirichlet : K
+=
+1000,  β
+=
+1, S
+true
+=
+0.839
+10
+2
+10
+3
+10
+4
+N
+10
+−2
+10
+−1
+RMSE
+10
+2
+10
+3
+10
+4 N
+−0.3
+−0.2
+−0.1
+0.0
+0.1
+ΔSrel
+Zipf : K = 1000,  a = 1.001, Strue = 0.751
+10
+2
+10
+3
+10
+4
+N
+10
+−2
+10
+−1
+RMSE
+10
+2
+10
+3
+10
+4
+N
+−0.1
+0.0
+0.1
+ΔS
+rel
+Bimodal : K
+=
+1000, S=0.857
+10
+2
+10
+3
+10
+4
+N
+10
+−2
+10
+−1
+RMSE
+FIG. 3. Shannon entropy estimation for atypical distributions in a
+Dirichlet prior (same legend as in Fig. 2): Dirichlet with β = 1
+but half bins are set to zeros; Zipf’s distribution with exponent a =
+1.001; bimodal distribution.
+
+6
+B.
+Kullback-Leibler divergence
+Regarding the Kullback-Leibler divergence DKL, there are
+no exact formulas for the moments of the posterior distribu-
+tion p(DKL|n). Therefore, we have to rely on a point estimate
+of the mean by first estimating the distributions via Laplace’s
+formula (5) with the inferred β⋆ and then plugging these val-
+ues into expression (8). The flat hyperprior in Eq. (13) is the
+only reasonable one to estimate β⋆ in this case, since the NSB
+prior (Eq. (7)) can only be justified for the Shannon entropy.
+10
+2
+10
+3
+10
+4 N
+0
+1
+2
+3
+DKL
+K = 1000,  β = 0.01
+β⋆
+plugin
+HS
+βplugin = ⋆
+10
+2
+10
+3
+10
+4 N
+0.00
+0.25
+0.50
+0.75
+1.00
+DKL
+K = 1000,  β = 1
+10
+2
+10
+3
+10
+4 N
+0.0
+0.2
+0.4
+0.6
+DKL
+K = 1000,  β = 10
+FIG. 4.
+Kullback-Leibler estimation for distributions typical in
+a Dirichlet prior, for β = 0.01, 1, 101 and sample size N
+=
+25 . . . 10000. Each point corresponds to an average over 1000 sam-
+ples. Black squares: our plug-in estimator, that is Laplace’s formula
+with β⋆ estimated from a flat hyperprior. Red crosses: Hausser-
+Strimmer plug-in estimator. Purple circles: Laplace’s estimator for
+uniform prior β = 1.
+We compare the results with Laplace’s estimator (5) with
+β = 1 and with the HS estimator, since both have the same
+desirable property of assigning non-null probabilities to un-
+observed states (ni = 0) and are suitable estimators for com-
+puting DKL. Indeed, β = 1 in Laplace’s formula is a com-
+mon choice and amounts to assigning the same probability
+to all possible distributions. We test the estimators in a sce-
+nario typical in machine learning and variational inference,
+in which one wants to minimize the DKL between a complex,
+target distribution and some model approximation. Here, after
+generating a synthetic discrete distribution ρ, we measure the
+DKL(ρ; ˆρ), where ˆρ is the distribution estimated from counts;
+hence a good estimator should make DKL as small as possi-
+ble.
+10
+2
+10
+3
+10
+4
+N
+0.0
+0.5
+1.0
+D
+KL
+Half empty Dirichlet : K
+=
+1000,  β
+=
+1
+10
+2
+10
+3
+10
+4
+N
+0.0
+0.5
+1.0
+D
+KL
+Zipf : K
+=
+1000,  a
+=
+1.001
+10
+2
+10
+3
+10
+4
+N
+0.0
+0.5
+1.0
+D
+KL
+Bimodal
+:
+ K
+=
+1000
+FIG. 5. Kullback-Leibler divergence estimation for atypical distribu-
+tions in a Dirichlet prior (same legend as in Fig. 4): Dirichlet with
+β = 1 but half bins set to zero; Zipf’s distribution with exponent
+a = 1, 001; bimodal distribution.
+In Figs. 4 and 5, we show that our estimator and the HS
+estimator provide similar results, although DKL(β⋆) is more
+
+7
+accurate in the very sparse regime N < 50, and when the tar-
+get distributions are atypical in the Dirichlet priors, especially
+in the important case of Zipf’s distributions. The estimator
+based on Laplace’s formula wih β = 1 performs generally
+worse, unless in the trivial case when the target distribution
+itself was also generated just from a Dirichlet with β = 1.
+Importantly, in this case in which β = 1 is optimal, our ap-
+proach provides virtually identical results.
+V.
+CONCLUSIONS
+Inferring the shape of discrete distributions and their infor-
+mation content from experimental data is a fundamental task
+in fields that spread from machine learning to computational
+social science and neuroscience. However, it is common in
+experiments to have a very low number of observations that
+hinder a correct estimation. In this paper, we have proposed
+a new method for the solution of this problem that applies
+to discrete distributions with a known number of states. It
+is pinned on the laws of conditional probability, in the form
+of Bayes’ rules, with the explicit assumption of a Dirichlet
+prior distribution as the mechanism behind the generation of
+data. In particular, we are able to provide a semi-analytical
+formula (Eq. (12)), easily solvable with moderated computa-
+tion efforts, to find the concentration parameter characterizing
+the Dirichlet distribution. This result is a step forward with re-
+spect to many previous works that share the same background
+but ultimately focused on constructing an infinite mixture of
+Dirichlet priors, which weights were chosen to optimize the
+estimation of the Shannon entropy only [12, 13, 18, 19]. Be-
+sides their precision and success, these other approaches are
+computationally involved and ignore any estimation of the
+probability distribution, which could be necessary, in partic-
+ular, for the estimation of the Kullback-Leibler divergence.
+Our approach allows the reconstruction of the posterior distri-
+bution of Shannon entropy for a broad variety of data types, by
+using the exact formulas in Ref. [16], with a precision compa-
+rable to or better than other estimators developed for the same
+purposes. In the case of Kullback-Leibler divergence, on the
+contrary, we were not able to estimate its full posterior dis-
+tribution, but we obtained a good point-wise estimation of its
+mean value, by estimating the two involved probability dis-
+tributions and then plugging them into the explicit expression
+of the Kulback-Leibler divergence. In regard to this point and
+for future studies, it is in general desirable having some ana-
+lytical expression for the posterior distribution (conditioned to
+observations) of the Kullback-Leibler divergence in the same
+spirit as the Shannon entropy. Further efforts should be de-
+voted to extending the same approach to more specific pri-
+ors than Dirichlet, for example for data that follow power
+law distributions, including Zipf’s laws, for binary distribu-
+tions [19], or when the number of states is unknown, as in
+Refs. [18, 20, 29, 30].
+VI.
+ACKNOWLEDGEMENTS
+This research was funded by the Social Observatory of the
+“la Caixa” Foundation as part of the project LCF / PR / SR19
+/ 52540009, by MCIN / AEI / 10.13039 / 501100011033
+(Project No. PID2019–106811GB-C31) and by the Govern-
+ment of Catalonia (Project No. 2017SGR-896).
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+Appendix A: Derivation of results (Eq. (12) in main text)
+Let us suppose that we have K different categories (or types of random events) and that we observe N independent random
+events distributed in the K categories n = {ni; i = 1, . . . , K}, with �
+i ni = N. We also assume that the probabilities of
+observing counts in each category ρi are distributed according to a Dirichlet prior with the same hyper-parameters β for all
+ρ = {ρi; i = 1, . . . , K}, so that
+p(ρ|β) =
+1
+BK(β)
+K
+�
+i=1
+ρβ−1
+i
+,
+BK(β) = Γ(β)K
+Γ(βK).
+(A1)
+Our goal is to compute the most likely value of β given the observed counts {ni}. To that end, we need to compute the
+conditional probability p(β|n). We can do this by marginalizing over the possible combinations of ρ = {ρi} as follows:
+p(β|n) = p(β)
+p(n)p(n|β) ,
+p(n|β) =
+�
+dρ p(n|β, ρ)p(ρ|β) .
+(A2)
+Since the probability of observing an event in category i is ρi, the probability of observing ni events of type i is ρni
+i . Therefore,
+for the integral in Eq. (A2) we have that
+p(n|β, ρ) =
+K
+�
+i=1
+ρni
+i
+,
+(A3)
+so that
+p(n|β) =
+1
+BK(β)
+�
+dρ
+K
+�
+i=1
+ρni+β−1
+i
+,
+(A4)
+where we have used Eq. (A1) for p(ρ|β) and the integral is over the simplex that satisfies the condition �
+i=1 ρi = 1.
+To perform the integrals above we first evaluate the normalization condition for ρk = 1 − R(K − 1) with RK−1 = �K−1
+i=1 ρi
+so that for ρk−1 we have the following integral:
+IK−1 =
+� 1−RK−2
+0
+dρK−1 ρnK−1+β−1
+K−1
+(1 − ρk−1 − RK−2)nK+β−1 .
+(A5)
+
+9
+To evaluate this integral we use the fact that
+� (1−R)
+0
+dx xa(1 − x − R)b = Γ(a + 1)Γ(b + 1)
+Γ(a + b + 2)
+(1 − R)a+b+1
+if
+Re(R) < 1
+and
+Im(R) = 0
+(A6)
+so that
+IK−1 = Γ(nK−1 + β)Γ(nK + β)
+Γ(nk + nK−1 + 2β)
+(1 − RK−2)nK+nK−1+2β−1
+(A7)
+Which gives for ρK−2 the following integral:
+IK−2 =
+� 1−RK−3
+0
+dρK−2 ρnK−2+β−1
+K−2
+(1 − ρK−2 − RK−3)nK+NK−1+2β−1
+(A8)
+= Γ(nK−2 + β)Γ(nK + nK−1 + 2β)
+Γ(nk + nK−1 + nk−2 + 3β)
+(1 − RK−3)nK+nK−1+nK−2+3β−1
+(A9)
+which have evaluated using Eq. (A6). If we do this for all ρ we end up having
+�
+dρ
+�
+i
+ρni+β−1
+i
+=
+K
+�
+i=1
+Ii =
+�K
+i=1 Γ(ni + β)
+Γ(N + Kβ)
+.
+(A10)
+Thus, we obtain the following expression for p(n|β)
+p(n|β) =
+1
+BK(β)
+�
+i Γ(ni + β)
+Γ(N + Kβ) = Γ(Kβ)
+Γ(β)K
+�
+i Γ(ni + β)
+Γ(N + Kβ)
+(A11)
+Our goal is to find β⋆ that maximizes p(β|n) = p(β)
+p(n)p(n|β). To that end we take the derivative of log p(β|n),
+log p(β|n) = log Γ(Kβ) − K log Γ(β) +
+�
+i
+log Γ(ni + β) − log Γ(N + Kβ) + log p(β) − log p(n)
+(A12)
+so that β⋆ is the one that satisfies the condition:
+d log p(β|n)
+dβ
+����
+β=β⋆ = 0 .
+(A13)
+To evaluate this equation we use the following definitions and properties of the log Gamma function:
+1.
+� d
+dx
+�m+1 log Γ(x) = ψm(x)
+(A14)
+2.
+ψ0(x + n) = �n−1
+m=0
+1
+x+m + ψ(x) .
+(A15)
+Using the expressions above and the consideration that p(β) = const. we obtain that:
+d log p(β|n)
+dβ
+= Kψ0(Kβ) − Kψ0(β) +
+�
+i
+ψ0(ni + β) − Kψ0(N + Kβ)
+(A16)
+=
+K
+�
+i=1
+ni−1
+�
+m=0
+1
+m + β −
+N−1
+�
+m=0
+K
+m + Kβ
+(A17)
+Therefore the condition that gives β⋆ is
+K
+�
+i=1
+ni−1
+�
+m=0
+1
+m + β⋆ −
+N−1
+�
+m=0
+K
+m + Kβ⋆ = 0 ,
+(A18)
+that is, the Eq. (12) in main text for uniform hyperprior (13). If instead we consider a prior for beta that results in a close-to-
+uniform distribution of Shannon entropy such as in Nemenman et al. [12, 13] then
+pNSB(β) = dS
+dβ ,
+(A19)
+
+10
+with S = E[S|ni = 0, β] = ψ0(Kβ + 1) − ψ0(β + 1), the average entropy of the distributions generated from a Dirichlet
+prior p(ρ|β). Note that this prior is already normalized since
+� ∞
+0
+dS/dβdβ = S(∞; K) − S(0; K) = 1. The derivative of the
+logarithm of this prior with respect to β is then
+d log pNSB(β)
+dβ
+=
+1
+pNSB(β)
+dpNSB(β)
+dβ
+= 1
+dS
+dβ
+d2S
+dβ2 = K2ψ2(kβ + 1) − ψ2(β + 1)
+Kψ1(kβ + 1) − ψ1(β + 1) ,
+which is the formula (12) in main text. The condition of the β⋆ that maximizes p(β|n) is in this case:
+d log p(β|n)
+dβ
+= Kψ0(Kβ) − Kψ0(β) +
+�
+i
+ψ0(ni + β) − Kψ0(N + Kβ) + 1
+dS
+dβ
+d2S
+dβ2 =
+=
+K
+�
+i=1
+ni−1
+�
+m=0
+1
+m + β⋆ −
+N−1
+�
+m=0
+K
+m + Kβ⋆ + K2ψ2(kβ⋆ + 1) − ψ2(β⋆ + 1)
+Kψ1(kβ⋆ + 1) − ψ1(β⋆ + 1) = 0 .
+Appendix B: Analytical moments of the Shannon entropy posterior
+In the specific case of S(ρ), instead of solving p (F|n) =
+�
+dρ δ (F − F(ρ)) p(ρ|n) (Eq. (2) in main text) directly, it is
+possible to obtain closed form expression for all the moments of the posterior [16–18]. Here we report the first two, the mean
+E[S|n, β] =
+�
+dρ S(ρ|β) p(ρ|n) = ψ0(N + Kβ + 1) −
+K
+�
+i=1
+ni + β
+N + Kβ ψ0(ni + β + 1) ,
+(B1)
+and the second moment
+E[S2|n, β] =
+�
+dρ S(ρ|β)2 p(ρ|n) =
+K
+�
+i̸=j
+(ni + β) (nj + β)
+(N + Kβ + 1) (N + Kβ) Ii,j +
+K
+�
+i=1
+(ni + β + 1) (ni + β)
+(N + Kβ + 1) (N + Kβ) Ji ,
+(B2)
+with
+Ii,j =
+�
+ψ0(ni + β + 1) − ψ0(N + Kβ + 2)
+�
+·
+�
+ψ0(nj + β + 1) − ψ0(N + Kβ + 2)
+�
+− ψ1(N + Kβ + 2) ;
+Ji =
+�
+ψ0(ni + β + 2) − ψ0(N + Kβ + 2)
+�2
++ ψ1(ni + β + 2) − ψ1(N + Kβ + 2) ;
+(B3)
+from which the standard deviation is in turn calculated as the square root of the variance Var(S|n, β) = E[S2|n, β]−E[S|n, β]2.
+