# A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamics

arXiv: https://arxiv.org/abs/2407.03945 · HTML: https://mlsanity.com/paper/2407.03945/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 35 references that are in the map.
- First posted on arXiv 4 Jul 2024.

## Builds on (most cited first, 8 of 8)

- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (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)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers](https://mlsanity.com/paper/2208.13273/) (2022)
- [Bounding the Rademacher Complexity of Fourier neural operators](https://mlsanity.com/paper/2209.05150/) (2022)
- [Effects of boundary conditions in fully convolutional networks for learning spatio-temporal dynamics](https://mlsanity.com/paper/2106.11160/) (2021)

## Built on by (most cited first, 4 of 4)

- [A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solvers](https://mlsanity.com/paper/2511.08811/) (2025)
- [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)
- [Mesh-Native Tokenisation: Retrofitting Foundation Models to Unstructured Data](https://mlsanity.com/paper/2610.12618/) (2026)

## Abstract (arXiv)

The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of larger time steps, and better structure preservation properties. However, this comes at the price of having to solve a nonlinear equation at every time step of the numerical scheme. In this work, we propose a novel deep learning based hybrid Newton's method to accelerate this solution of the nonlinear time step system for stiff time-evolution nonlinear equations. We propose a targeted learning strategy which facilitates robust unsupervised learning in an offline phase and provides a highly efficient initialisation for the Newton iteration leading to consistent acceleration of Newton's method. A quantifiable rate of improvement in Newton's method achieved by improved initialisation is provided and we analyse the upper bound of the generalisation error of our unsupervised learning strategy. These theoretical results are supported by extensive numerical results, demonstrating the efficiency of our proposed neural hybrid solver both in one- and two-dimensional cases.