mlsanityData to 14 Oct 2026

A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solvers

arXiv 2511.08811 ↗First posted 11 Nov 2025math.NA cs.LG cs.NAcited by 5

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Abstract (arXiv)

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.

Key facts. Cited by 5 papers in the map (data through 14 Oct 2026). Builds on 19 of its 49 references that are in the map. First posted on arXiv 11 Nov 2025.

Builds on

19 of 49 references.

References · 19

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7 more
  1. 2022Blending Neural Operators and Relaxation Methods in PDE Numerical Solverscited by 54
  2. 2024Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularitycited by 34
  3. 2022A Deep Conjugate Direction Method for Iteratively Solving Linear Systemscited by 12
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 5.

5 papers.

  1. 2026Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solverscited by 1
  2. 2026Twincher: Bijective Representation Learning for Robust Inversion of Continuous Systemscited by 0
  3. 2026Learning features from Newton's algorithm: a way to accelerate nonlinear parametrized PDE solverscited by 0
  4. 2026An overview of machine learning-enhanced iterative methods for systems of linear and nonlinear equationscited by 0
  5. 2026Transferability of Learned States in Neural PDE Solverscited by 0