A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamics

TL;DR

This paper introduces a deep learning-based hybrid Newton's method to efficiently solve nonlinear equations in implicit schemes for stiff nonlinear time-evolution equations, improving convergence speed and stability.

Contribution

It presents a novel neural hybrid Newton solver with an unsupervised learning strategy for better initialisation, accelerating nonlinear system solutions in implicit time-stepping.

Findings

Achieves significant acceleration of Newton's method in numerical experiments.

Demonstrates robustness and efficiency in 1D and 2D nonlinear dynamics problems.

Provides theoretical analysis of generalisation error bounds.

Abstract

The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of larger time steps, and better structure preservation properties. However, this comes at the price of having to solve a nonlinear equation at every time step of the numerical scheme. In this work, we propose a novel deep learning based hybrid Newton's method to accelerate this solution of the nonlinear time step system for stiff time-evolution nonlinear equations. We propose a targeted learning strategy which facilitates robust unsupervised learning in an offline phase and provides a highly efficient initialisation for the Newton iteration leading to consistent acceleration of Newton's method. A quantifiable rate of improvement in Newton's method achieved by improved initialisation is provided and we analyse the upper bound of the generalisation error of our unsupervised learning strategy. These theoretical results are supported by extensive numerical results, demonstrating the efficiency of our proposed neural hybrid solver both in one- and two-dimensional cases.