shiwen0710/Heat-Gauss-Lshape

Dataset Description

Overview

This dataset contains solutions to the two-dimensional parametric heat equation given by:

$$u_t=\mathrm{\Delta }u,t\in [0,T],(x_1,x_2)\in [0,1{]}^2,\mu \in [-1,1{]}^{d}u\_t\; =\; \backslash Delta\; u,\; \backslash quad\; t\; \backslash in\; [0,\; T],\; \backslash left(x\_1,\; x\_2\backslash right)\; \backslash in\; [0,1]^2,\; \backslash mu\; \backslash in\; [-1,1]^d$$

with the initial condition:

$$\begin{array}{c}{\displaystyle u(0,x_1,x_2,\mu )=u_0(x_1,x_2,\mu )=\frac{1}{d}\sum _{m=1}^d{u}_0^m(x_1,x_2,{\mu }_m)=-\frac{1}{d}\sum _{m=1}^d{\mu }_m\sin \left(\pi m{x}_1\right)\sin \left(\pi m{x}_2\right)\mathrm{/}{m}^{0.5}}\\ {\displaystyle u_0^m(x_1,x_2,{\mu }_m)=-{\mu }_m\sin \left(\pi m{x}_1\right)\sin \left(\pi m{x}_2\right)\mathrm{/}{m}^{0.5}}\end{array}\backslash begin\{gathered\}\; u\backslash left(0,\; x\_1,\; x\_2,\; \backslash mu\backslash right)\; =\; u\_0\backslash left(x\_1,\; x\_2,\; \backslash mu\backslash right)\; =\; \backslash frac\{1\}\{d\}\; \backslash sum\_\{m=1\}^d\; u\_0^m\backslash left(x\_1,\; x\_2,\; \backslash mu\_m\backslash right)\; =\; -\backslash frac\{1\}\{d\}\; \backslash sum\_\{m=1\}^d\; \backslash mu\_m\; \backslash sin\; \backslash left(\backslash pi\; m\; x\_1\backslash right)\; \backslash sin\; \backslash left(\backslash pi\; m\; x\_2\backslash right)\; /\; m^\{0.5\}\; \backslash \backslash \; u\_0^m\backslash left(x\_1,\; x\_2,\; \backslash mu\_m\backslash right)\; =\; -\backslash mu\_m\; \backslash sin\; \backslash left(\backslash pi\; m\; x\_1\backslash right)\; \backslash sin\; \backslash left(\backslash pi\; m\; x\_2\backslash right)\; /\; m^\{0.5\}\; \backslash end\{gathered\}$$

Dataset Specifications

In this dataset, the parameter dimension is set to $d = 16$. The dataset contains 5000 sampled trajectories for $\mu \sim \operatorname{Unif}([-1,1]^d)$ within an L-shaped domain. We impose zero Dirichlet boundary conditions.

The solution was computed on a triangular mesh generated using the Delaunay algorithm with the following specifications:

• Number of points: 14047
• Mesh elements:
  • Lines: 502
  • Triangles: 27590

Temporal discretization is performed with a time step $\Delta t = 0.00005$, and each trajectory is simulated for 100 time steps.

Data Structure

Each dataset file (e.g., heat_dataset_2000_3000.npz) contains the following arrays:

• x: (Array of shape [14047,]) The x-coordinates of the mesh points.
• y: (Array of shape [14047,]) The y-coordinates of the mesh points.
• u0: (Array of shape [14047,]) The initial condition of the solution at time ( t = 0 ).
• u: (Array of shape [1000, 100, 14047]) The solution at each time step for each sampled trajectory:
  • 1000: Number of trajectories.
  • 100: Number of time steps per trajectory.
  • 14047: Mesh points per solution snapshot.
• dt: (Float) The time step ( \Delta t = 0.00005 ).

Applications

This dataset is suitable for applications in:

• Reduced-order modeling
• Neural Operator
• Machine learning for scientific computing
• Parametric PDE solvers

Additional Information

• The dataset is constructed using finite element methods (TensorMesh) on an L-shaped domain.
• Simulated time: ~270s/1000 trajectories.