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

TL;DR

The paper tackles the computational burden of solving nonlinear PDEs in solid mechanics by introducing Neural-Initialized Newton (NiN), a hybrid approach that initializes the Newton–Raphson solver with predictions from a physics-informed neural operator (iFOL). By leveraging zero-shot super-resolution and a PDE-residual–guided neural field, NiN preserves NFEM accuracy while dramatically reducing iterations and compute time across 2D and 3D hyperelasticity and multiphysics problems. The key contributions include designing a neural-warm-start strategy that can be applied to complex geometries and high-fidelity simulations, demonstrating substantial speed-ups (often 1–2 orders of magnitude) without sacrificing accuracy, and providing a flexible framework compatible with multiple nonlinear solvers. The results highlight the practical impact for large-scale nonlinear simulations, design optimization, and digital-twin workflows, with an open-source implementation to enable broader adoption.

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.

Figures (22)

• Figure 1: Schematic representation of the three methods investigated in this work.
• Figure 2: Nonlinear problems investigated in this work. Left: learning deformation patterns for arbitrary material distributions. Middle: learning solution fields on complex geometries under arbitrary loading conditions. Right: learning multiphysics solution fields within a representative volume element.
• Figure 3: Architecture for physics-informed operator learning to map property fields to solution fields: training on random low-fidelity Fourier samples (left) and inference on realistic high-resolution microstructures (right).
• Figure 4: Illustration of a few samples from the training (first row) and test (second row) datasets. Note that during training, the morphologies are limited to low-resolution, Fourier-based, symmetric periodic structures with constant phase contrast, whereas in the test cases we increase the resolution, vary the topologies, and adjust the phase contrast values for more challenging scenarios.
• Figure 5: Proposed architecture to map applied Dirichlet boundary conditions to solution fields: training on random loading values (left) and inference on structural responses with complex geometries (right).