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

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.