mlsanityData to 7 Oct 2026

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

arXiv 2511.06802 ↗First posted 10 Nov 2025cs.LGcited by 6

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Abstract (arXiv)

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.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 35 of its 63 references that are in the map. First posted on arXiv 10 Nov 2025.

Builds on

35 of 63 references.

References · 35

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
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23 more
  1. 2021Improved architectures and training algorithms for deep operator networkscited by 85
  2. 2022Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modelingcited by 65
  3. 2022Blending Neural Operators and Relaxation Methods in PDE Numerical Solverscited by 53
  4. 2022A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element methodcited by 33
  5. 2023Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loadscited by 33
  6. 2023Rethinking materials simulations: Blending direct numerical simulations with neural operatorscited by 25
  7. 2022Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problemscited by 21
  8. 2021Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligencecited by 17
  9. 2024A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domainscited by 15
  10. 2022Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanicscited by 13
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  12. 2024A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformationscited by 10
  13. 2023Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problemscited by 9
  14. 2024A Spectral-based Physics-informed Finite Operator Learning for Prediction of Mechanical Behavior of Microstructurescited by 9
  15. 2025NOWS: Neural Operator Warm Starts for Accelerating Iterative Solverscited by 9
  16. 2025PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulationscited by 8
  17. 2023Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINNcited by 6
  18. 2025Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Methodcited by 6
  19. 2025A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometriescited by 5
  20. 2024A spatiotemporal deep learning framework for prediction of crack dynamics in heterogeneous solids: efficient mapping of concrete microstructures to its fracture propertiescited by 3
  21. 2025EquiNO: A Physics-Informed Neural Operator for Multiscale Simulationscited by 3
  22. 2024Finite Operator Learning: Bridging Neural Operators and Numerical Methods for Efficient Parametric Solution and Optimization of PDEscited by 2
  23. 2024Finite-PINN: A Physics-Informed Neural Network with Finite Geometric Encoding for Solid Mechanicscited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 6.

6 papers.

  1. 2026Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solverscited by 1
  2. 2026NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equationscited by 0
  3. 2026Tackling multiphysics problems via finite element-guided physics-informed operator learningcited by 0
  4. 2026SurgFormer: Scalable Learning of Organ Deformation with Resection Support and Real-Time Inferencecited by 0
  5. 2026Interface-Aware Neural Newton Preconditioning for Robust Cohesive Zone Model Simulationscited by 0
  6. 2026Learning features from Newton's algorithm: a way to accelerate nonlinear parametrized PDE solverscited by 0