A Review of Neural Network Solvers for Second-order Boundary Value Problems

TL;DR

This paper reviews recent deep learning-based neural network methods for solving second-order boundary value problems, comparing their performance, analyzing errors, and discussing limitations in the field.

Contribution

It provides a comprehensive comparison and analysis of several neural network PDE solvers like PINN, WAN, DRM, and VPINN, highlighting their differences and challenges.

Findings

PINN error analysis reveals key sources of inaccuracies

Loss formulation significantly impacts solver performance

Optimization methods influence convergence and accuracy

Abstract

Deep learning-based partial differential equation(PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into highly nonlinear optimization problems of neural network parameters. This work reviews several deep learning solvers proposed a few years ago, including PINN, WAN, DRM, and VPINN. Numerical results are provided to make comparisons amongst them and address the importance of loss formulation and the optimization method. A rigorous error analysis for PINN is also presented. Finally, we discuss the current limitations and bottlenecks of these methods.