Newton Informed Neural Operator for Computing Multiple Solutions of Nonlinear Partials Differential Equations

TL;DR

This paper introduces a Newton Informed Neural Operator that effectively learns multiple solutions of nonlinear PDEs, overcoming limitations of traditional neural network methods and reducing data requirements.

Contribution

It presents a novel neural operator that integrates Newton methods to handle multiple solutions and improve efficiency in solving nonlinear PDEs.

Findings

Learns multiple solutions in a single training process

Requires fewer supervised data points than existing methods

Successfully addresses ill-posedness in nonlinear PDEs

Abstract

Solving nonlinear partial differential equations (PDEs) with multiple solutions using neural networks has found widespread applications in various fields such as physics, biology, and engineering. However, classical neural network methods for solving nonlinear PDEs, such as Physics-Informed Neural Networks (PINN), Deep Ritz methods, and DeepONet, often encounter challenges when confronted with the presence of multiple solutions inherent in the nonlinear problem. These methods may encounter ill-posedness issues. In this paper, we propose a novel approach called the Newton Informed Neural Operator, which builds upon existing neural network techniques to tackle nonlinearities. Our method combines classical Newton methods, addressing well-posed problems, and efficiently learns multiple solutions in a single learning process while requiring fewer supervised data points compared to existing neural network methods.