Weak TransNet: A Petrov-Galerkin based neural network method for solving elliptic PDEs

TL;DR

Weak TransNet introduces a Petrov-Galerkin neural network approach for solving elliptic PDEs, effectively handling solutions with low regularity or singularities, and demonstrating robustness and efficiency through numerical experiments.

Contribution

The paper proposes the Weak TransNet method, combining neural networks with a Petrov-Galerkin formulation to improve solving elliptic PDEs, especially with complex solution features.

Findings

Effective in handling low regularity solutions

Mitigates neural network training issues like non-convexity

Demonstrates robustness and efficiency in numerical tests

Abstract

While deep learning has achieved remarkable success in solving partial differential equations (PDEs), it still faces significant challenges, particularly when the PDE solutions have low regularity or singularities. To address these issues, we propose the Weak TransNet (WTN) method, based on a Petrov-Galerkin formulation, for solving elliptic PDEs in this work, though its framework may extend to other classes of equations. Specifically, the neural feature space defined by TransNet (Zhang et al., 2023) is used as the trial space, while the test space is composed of radial basis functions. Since the solution is expressed as a linear combination of trial functions, the coefficients can be determined by minimizing the weak PDE residual via least squares. Thus, this approach could help mitigate the challenges of non-convexity and ill-conditioning that often arise in neural network training. Furthermore, the WTN method is extended to handle problems whose solutions exhibit multiscale features or possess sharp variations. Several numerical experiments are presented to demonstrate the robustness and efficiency of the proposed methods.