# NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers

arXiv: https://arxiv.org/abs/2511.02481 · HTML: https://mlsanity.com/paper/2511.02481/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 53 references that are in the map.
- First posted on arXiv 4 Nov 2025.

## Builds on (most cited first, 21 of 21)

- [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)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers](https://mlsanity.com/paper/2208.13273/) (2022)
- [MgNet: A Unified Framework of Multigrid and Convolutional Neural Network](https://mlsanity.com/paper/1901.10415/) (2019)
- [Equivariant Graph Neural Operator for Modeling 3D Dynamics](https://mlsanity.com/paper/2401.11037/) (2024)
- [Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularity](https://mlsanity.com/paper/2405.19166/) (2024)
- [Multigrid-augmented deep learning preconditioners for the Helmholtz equation](https://mlsanity.com/paper/2203.11025/) (2022)
- [Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problem](https://mlsanity.com/paper/2405.12380/) (2024)
- [UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs](https://mlsanity.com/paper/2408.04846/) (2024)
- [Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method](https://mlsanity.com/paper/2508.00101/) (2025)
- [AI-enhanced iterative solvers for accelerating the solution of large scale parametrized systems](https://mlsanity.com/paper/2207.02543/) (2022)
- [Reducing operator complexity in Algebraic Multigrid with Machine Learning Approaches](https://mlsanity.com/paper/2307.07695/) (2023)
- [Active Learning with Bayesian Multi-Fidelity Laplace Neural Operators for Oscillatory Parametric PDEs](https://mlsanity.com/paper/2502.00550/) (2025)
- [Accelerating Conjugate Gradient Solvers for Homogenization Problems with Unitary Neural Operators](https://mlsanity.com/paper/2508.02681/) (2025)

## Built on by (most cited first, 9 of 9)

- [Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learning](https://mlsanity.com/paper/2511.06802/) (2025)
- [WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains](https://mlsanity.com/paper/2605.24651/) (2026)
- [Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solvers](https://mlsanity.com/paper/2606.21828/) (2026)
- [Learning Physics from an Imperfect Ancestor](https://mlsanity.com/paper/2609.24947/) (2026)
- [HAMNO: A Hierarchical Adaptive Multi-scale Neural Operator with Physics-Informed Learning for Dynamical Systems](https://mlsanity.com/paper/2606.11963/) (2026)
- [GA-VINO: A Geometry-Aware Variational Physics-informed Neural Operator for Mindlin-Reissner Plates](https://mlsanity.com/paper/2606.16624/) (2026)
- [Interface-Aware Neural Newton Preconditioning for Robust Cohesive Zone Model Simulations](https://mlsanity.com/paper/2606.31921/) (2026)
- [Neural Operators for Immersed-Boundary Soft Swimmers Locomotion](https://mlsanity.com/paper/2608.07722/) (2026)
- [HiLNO: A Hierarchical Latent Neural Operator with Multi-Scale Supervision for PDEs on General Geometries](https://mlsanity.com/paper/2609.18419/) (2026)

## Abstract (arXiv)

Partial differential equations (PDEs) underpin quantitative descriptions across the physical sciences and engineering, yet high-fidelity simulation remains a major computational bottleneck for many-query, real-time, and design tasks. Data-driven surrogates can be strikingly fast but are often unreliable when applied outside their training distribution. Here we introduce Neural Operator Warm Starts (NOWS), a hybrid strategy that harnesses learned solution operators to accelerate classical iterative solvers by producing high-quality initial guesses for Krylov methods such as conjugate gradient and GMRES. NOWS leaves existing discretizations and solver infrastructures intact, integrating seamlessly with finite-difference, finite-element, isogeometric analysis, finite volume method, etc. Across our benchmarks, the learned initialization consistently reduces iteration counts and end-to-end runtime, resulting in a reduction of the computational time of up to 90 %, while preserving the stability and convergence guarantees of the underlying numerical algorithms. By combining the rapid inference of neural operators with the rigor of traditional solvers, NOWS provides a practical and trustworthy approach to accelerate high-fidelity PDE simulations.