A Comparative Study of Neural Network Solvers for Second‐Order Boundary Value Problems

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 does a survey and comparative study of several deep‐learning solvers proposed a few years back, including PINN, WAN, DRM, and VPINN. Numerical results are provided to make comparisons among them and address the importance of loss formulation and the optimization method. A rigorous error analysis for PINN is presented, and brief error estimate results for other methods are discussed. Finally, we discuss the current limitations and bottlenecks of these methods.