DeepONet for Solving Nonlinear Partial Differential Equations with Physics-Informed Training

TL;DR

This paper explores the use of DeepONet, a neural operator learning method, for solving nonlinear PDEs with physics-informed training, providing theoretical error bounds and demonstrating performance improvements.

Contribution

It introduces a rigorous error analysis for DeepONet in physics-informed PDE solving, highlighting the importance of network architecture choices and deriving generalization bounds.

Findings

Complex branch networks improve performance

Simpler trunk networks are more effective

Derived a bound on generalization error in Sobolev norms

Abstract

In this paper, we investigate the applications of operator learning, specifically DeepONet, for solving nonlinear partial differential equations (PDEs). Unlike conventional function learning methods that require training separate neural networks for each PDE, operator learning enables generalization across different PDEs without retraining. This study examines the performance of DeepONet in physics-informed training, focusing on two key aspects: (1) the approximation capabilities of deep branch and trunk networks, and (2) the generalization error in Sobolev norms. Our results show that complex branch networks provide substantial performance gains, while trunk networks are most effective when kept relatively simple. Furthermore, we derive a bound on the generalization error of DeepONet for solving nonlinear PDEs by analyzing the Rademacher complexity of its derivatives in terms of pseudo-dimension. This work bridges a critical theoretical gap by delivering rigorous error estimates. This paper fills a theoretical gap by providing error estimates for a wide range of physics-informed machine learning models and applications.