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

International Journal for Numerical Methods in Engineering, vol. 127

Abstract

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 neural operator based on conditional neural fields or Fourier neural operators is trained to approximate the nonlinear parametric solution of 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. This effect becomes even more pronounced when correcting solutions at significantly higher resolutions, effectively enabling a reduced‐order modelling strategy during the training phase without loss of accuracy after inference. The conducted studies are intentionally designed to cover a broad spectrum of problems, including variations in material properties, boundary conditions, complex geometries, and nonlinear multi‐physics scenarios, thereby demonstrating the robustness and general applicability of the approach.

Authors 5

1. Kianoosh Taghikhani corresponding Aachen Access e.V Aachen Germany Institute of Applied Mechanics

   Access · RWTH Aachen University

   Affiliation as printed

   ACCESS e.V. Aachen Germany

   Institute of Applied Mechanics RWTH Aachen University Aachen Germany

2. Yusuke Yamazaki

   Keio University

   Affiliation as printed

   Graduate School of Science and Technology Keio University Kohokuku, Yokohama Japan

3. Markus Apel Aachen Access e.V Aachen Germany

   Access

   Affiliation as printed

   ACCESS e.V. Aachen Germany

4. Reza Najian Asl

   Technical University of Munich

   Affiliation as printed

   Technical University of Munich Munich Germany

5. Shahed Rezaei corresponding Aachen Access e.V Aachen Germany

   Access

   Affiliation as printed

   ACCESS e.V. Aachen Germany