Newton Informed Neural Operator for Solving Nonlinear Partial Differential Equations

Wenrui Hao, Xinliang Liu, Yahong Yang

Keywords: operator Learning, nonlinear partial differential equations, multiple solutions, Newton method

TL;DR: We develop a Newton-informed neural operator for solving nonlinear problems with multiple solutions, requiring less data and offering faster convergence than traditional Newton methods.

Abstract:

Solving nonlinear partial differential equations (PDEs) with multiple solutions is essential in various fields, including physics, biology, and engineering. However, traditional numerical methods, such as finite element and finite difference methods, often face challenges when dealing with nonlinear solvers, particularly in the presence of multiple solutions. These methods can become computationally expensive, especially when relying on solvers like Newton's method, which may struggle with ill-posedness near bifurcation points.
In this paper, we propose a novel approach, the Newton Informed Neural Operator, which learns the Newton solver for nonlinear PDEs. Our method integrates traditional numerical techniques with the Newton nonlinear solver, efficiently learning the nonlinear mapping at each iteration. This approach allows us to compute multiple solutions in a single learning process while requiring fewer supervised data points than existing neural network methods.

Primary Area: Machine learning for physical sciences (for example: climate, physics)

Submission Number: 7292