A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solvers

TL;DR

This paper introduces a neural preconditioned Newton method utilizing a neural operator to improve convergence and robustness in solving complex parametric nonlinear systems, especially with strong nonlinearities.

Contribution

It presents a novel neural preconditioning approach with a fixed-point neural operator that adaptively enhances Newton methods for nonlinear solvers.

Findings

Demonstrates improved computational efficiency in real-world applications.

Shows robustness against unbalanced and strong nonlinearities.

Outperforms traditional methods in stability and convergence speed.

Abstract

We propose a novel neural preconditioned Newton (NP-Newton) method for solving parametric nonlinear systems of equations. To overcome the stagnation or instability of Newton iterations caused by unbalanced nonlinearities, we introduce a fixed-point neural operator (FPNO) that learns the direct mapping from the current iterate to the solution by emulating fixed-point iterations. Unlike traditional line-search or trust-region algorithms, the proposed FPNO adaptively employs negative step sizes to effectively mitigate the effects of unbalanced nonlinearities. Through numerical experiments we demonstrate the computational efficiency and robustness of the proposed NP-Newton method across multiple real-world applications, especially for very strong nonlinearities.