Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEs

TL;DR

This paper proves the convergence of an unsupervised spectral neural network method for solving linear second-order elliptic PDEs, combining deep learning with spectral Galerkin techniques.

Contribution

It introduces ULGNet, a novel spectral neural network approach that expresses solutions via Legendre basis and proves its convergence to PDE solutions.

Findings

The discrete loss minimizer converges to the weak PDE solution.

Numerical experiments support the theoretical convergence results.

The method effectively combines spectral methods with deep learning.

Abstract

In this paper, we perform the convergence analysis of unsupervised Legendre--Galerkin neural networks (ULGNet), a deep-learning-based numerical method for solving partial differential equations (PDEs). Unlike existing deep learning-based numerical methods for PDEs, the ULGNet expresses the solution as a spectral expansion with respect to the Legendre basis and predicts the coefficients with deep neural networks by solving a variational residual minimization problem. Since the corresponding loss function is equivalent to the residual induced by the linear algebraic system depending on the choice of basis functions, we prove that the minimizer of the discrete loss function converges to the weak solution of the PDEs. Numerical evidence will also be provided to support the theoretical result. Key technical tools include the variant of the universal approximation theorem for bounded neural networks, the analysis of the stiffness and mass matrices, and the uniform law of large numbers in terms of the Rademacher complexity.