Subspace method based on neural networks for solving the partial differential equation in weak form

TL;DR

The paper tackles efficient solutions of PDEs in weak form by constructing a subspace $V_h=\mathrm{span}\{\varphi_j\}_{j=1}^M$ from neural-network-based base functions and solving the weak form via Galerkin: $u_h=\sum_{j=1}^M \omega_j\varphi_j$, with $A_{ij}=a(\varphi_j,\varphi_i)$ and $b_i=(f,\varphi_i)$. Base functions are trained separately using flexible loss formulations corresponding to PINN, DGM, or DRM strategies, enabling either PDE-loss or bilinear-form loss, and the final step is a linear system in $\omega$. The method achieves high accuracy (down to $||e||_{L^2}<10^{-7}$ in some tests) with relatively few training epochs (hundreds to a few thousand) and substantially lower cost than standard weak-form neural PDE solvers. Validations on Helmholtz, Poisson, and anisotropic diffusion demonstrate superior accuracy-cost performance and robustness to strong anisotropy, highlighting practical impact for low-to-moderate dimensional PDEs.

Abstract

We present a subspace method based on neural networks for solving the partial differential equation in weak form with high accuracy. The basic idea of our method is to use some functions based on neural networks as base functions to span a subspace, then find an approximate solution in this subspace. Training base functions and finding an approximate solution can be separated, that is different methods can be used to train these base functions, and different methods can also be used to find an approximate solution. In this paper, we find an approximate solution of the partial differential equation in the weak form. Our method can achieve high accuracy with low cost of training. Numerical examples show that the cost of training these base functions is low, and only one hundred to two thousand epochs are needed for most tests. The error of our method can fall below the level of $10^{-7}$ for some tests. The proposed method has the better performance in terms of the accuracy and computational cost.

Figures (10)

• Figure 1: The neural networks architecture.
• Figure 2: Solution obtained by SNNW for Helmholtz Equation.
• Figure 3: Point-wise errors of SNNW-P, SNNW-G and SNNW-R for Helmholtz equation.
• Figure 4: Error variation with subspace dimension at a fixed number of 1000 sampling points and error variation with the number of sampling points at a fixed subspace dimension of 300 for SNNW-P.
• Figure 5: Error variation with subspace dimension at a fixed number of 1000 sampling points and error variation with the number of sampling points at a fixed subspace dimension of 300 for SNNW-G.