A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solvers

TL;DR

This work introduces NP-Newton, a neural preconditioned Newton method that employs a Fixed Point Neural Operator (FPNO) as a nonlinear right preconditioner to alleviate stagnation from unbalanced nonlinearities in parametric nonlinear systems $\pazocal{F}(u)=0$. The FPNO learns a fixed-point map and permits negative step sizes, enabling robust, fast convergence even in challenging regimes, with the MIONet backbone enabling efficient handling of parametric PDEs. Across nonlinear Poisson and hyper elasticity benchmarks, NP-Newton significantly reduces iteration counts and wall-clock time compared with standard Newton variants and incremental loading, demonstrating strong robustness and practical impact for real-time or large-scale nonlinear solves. The framework opens pathways to integrate data-driven preconditioning with established domain-decomposition and multilevel strategies to tackle large-scale problems in computational mechanics and physics-informed simulations.

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.

Figures (5)

• Figure 1: A schematic of neural preconditioned Newton method using the fixed point neural operator (FPNO) with backbone neural operator $\pazocal{G}_{B}$. Starting from the initial guess $u^{(0)}$, the FPNO is first applied to get the smoothed iterate $v^{(i)}$ and the Newton's method is applied to obtain the next iterate $u^{(i+1)}$.
• Figure 1: The numerical solution of nonlinear Poisson equation \ref{['eqn:fem_nonlinear_poisson']} computed on $129 \times 129$ mesh, where $q(u)=0.01+u^{2}$ and $f(x,y)=\sin(\pi x)\sin(\pi y)$.
• Figure 1: The convergence of the Newton's method for the nonlinear Poisson equation. The first row shows the results for the coarse mesh, and the second row shows the results for the fine mesh. In Case III, the Newton-LS method diverges, and the iteration is terminated when the relative residual norm exceeds $10^{4}$.
• Figure 2: The computed displacement of hyper elasticity problem \ref{['eqn:hyper_elasticity']} in $\Omega \subset (0,1)^{2}$ when $u_{t}=1$ and the maximum mesh size $h = 1/32$. Note that the gray object is the initial mesh.
• Figure 2: Convergence of Newton's method for the hyper elasticity problem. Since the IC-Newton-LS uses the incremental loading technique, the history of the relative residual norm is concatenated. In Case I, the convergence behaviors of NP-Newton-LS and NP-Newton-TR are almost identical, so their plots overlap.

Theorems & Definitions (1)

• Remark 2.1