Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learning

TL;DR

This paper introduces a hybrid method combining neural operator predictions with Newton's method to efficiently and accurately solve nonlinear parametric problems in computational solid mechanics, reducing computational costs.

Contribution

It presents a neural-initialized Newton approach that accelerates nonlinear finite element solutions by integrating neural operator predictions with traditional Newton refinement.

Findings

Reduces computational cost compared to standard Newton methods.

Maintains high accuracy while improving efficiency.

Outperforms pure neural operator approaches outside training distribution.

Abstract

We propose a Newton-based scheme, initialized by neural operator predictions, to accelerate the parametric solution of nonlinear problems in computational solid mechanics. First, a physics informed conditional neural field is trained to approximate the nonlinear parametric solutionof the governing equations. This establishes a continuous mapping between the parameter and solution spaces, which can then be evaluated for a given parameter at any spatial resolution. Second, since the neural approximation may not be exact, it is subsequently refined using a Newton-based correction initialized by the neural output. To evaluate the effectiveness of this hybrid approach, we compare three solution strategies: (i) the standard Newton-Raphson solver used in NFEM, which is robust and accurate but computationally demanding; (ii) physics-informed neural operators, which provide rapid inference but may lose accuracy outside the training distribution and resolution; and (iii) the neural-initialized Newton (NiN) strategy, which combines the efficiency of neural operators with the robustness of NFEM. The results demonstrate that the proposed hybrid approach reduces computational cost while preserving accuracy, highlighting its potential to accelerate large-scale nonlinear simulations.