A Complete Error Analysis of PINNs for Elliptic Equations Using Projected Stochastic Gradient Descent

Abstract

Physics-informed neural networks (PINNs) have recently gained attention as a powerful and efficient tool for solving partial differential equations (PDEs). Despite their empirical success, the theoretical understanding of PINNs, especially in the context of over-parameterization, remains incomplete. This paper presents a complete error analysis of over-parameterized PINNs for elliptic equations using projected stochastic gradient descent (PSGD) optimization. Our analysis rigorously examines the interplay of approximation error, statistical error, and optimization error, offering a unified framework for understanding the convergence behavior of PINNs. By leveraging the properties of PSGD, we establish convergence rates and derive conditions on neural network architecture, training sample requirements, and optimization parameters to ensure specified accuracy.