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

Abstract: In this paper, we investigate the use 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 demonstrate that deep branch networks provide substantial performance improvements, while trunk networks achieve optimal results 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 estimations for a wide range of physics-informed machine learning models and applications.