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

arXiv: https://arxiv.org/abs/2511.06802 · HTML: https://mlsanity.com/paper/2511.06802/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 63 references that are in the map.
- First posted on arXiv 10 Nov 2025.

## Builds on (most cited first, 30 of 35)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers](https://mlsanity.com/paper/2208.13273/) (2022)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads](https://mlsanity.com/paper/2306.03645/) (2023)
- [Rethinking materials simulations: Blending direct numerical simulations with neural operators](https://mlsanity.com/paper/2312.05410/) (2023)
- [Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2210.03008/) (2022)
- [Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligence](https://mlsanity.com/paper/2108.02837/) (2021)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics](https://mlsanity.com/paper/2207.09908/) (2022)
- [FEANN - An efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining](https://mlsanity.com/paper/2207.01045/) (2022)
- [A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations](https://mlsanity.com/paper/2404.00074/) (2024)
- [Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems](https://mlsanity.com/paper/2306.12047/) (2023)
- [A Spectral-based Physics-informed Finite Operator Learning for Prediction of Mechanical Behavior of Microstructures](https://mlsanity.com/paper/2410.19027/) (2024)
- [NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers](https://mlsanity.com/paper/2511.02481/) (2025)
- [PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulations](https://mlsanity.com/paper/2507.02227/) (2025)
- [Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINN](https://mlsanity.com/paper/2304.06044/) (2023)
- [Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method](https://mlsanity.com/paper/2508.00101/) (2025)

## Built on by (most cited first, 7 of 7)

- [Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solvers](https://mlsanity.com/paper/2606.21828/) (2026)
- [NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equations](https://mlsanity.com/paper/2601.17207/) (2026)
- [Tackling multiphysics problems via finite element-guided physics-informed operator learning](https://mlsanity.com/paper/2603.01420/) (2026)
- [SurgFormer: Scalable Learning of Organ Deformation with Resection Support and Real-Time Inference](https://mlsanity.com/paper/2603.06543/) (2026)
- [Interface-Aware Neural Newton Preconditioning for Robust Cohesive Zone Model Simulations](https://mlsanity.com/paper/2606.31921/) (2026)
- [Learning features from Newton's algorithm: a way to accelerate nonlinear parametrized PDE solvers](https://mlsanity.com/paper/2607.28036/) (2026)
- [An overview of machine learning-enhanced iterative methods for systems of linear and nonlinear equations](https://mlsanity.com/paper/2610.07211/) (2026)

## 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.