# Accurate and scalable deep Maxwell solvers using multilevel iterative methods

arXiv: https://arxiv.org/abs/2509.03622 · HTML: https://mlsanity.com/paper/2509.03622/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 50 references that are in the map.
- First posted on arXiv 3 Sep 2025.

## Builds on (most cited first, 19 of 19)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [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)
- [Deep neural networks for the evaluation and design of photonic devices](https://mlsanity.com/paper/2007.00084/) (2020)
- [Synthetic Lagrangian Turbulence by Generative Diffusion Models](https://mlsanity.com/paper/2307.08529/) (2023)
- [Global optimization of dielectric metasurfaces using a physics-driven neural network](https://mlsanity.com/paper/1906.04157/) (2019)
- [Multigrid-augmented deep learning preconditioners for the Helmholtz equation](https://mlsanity.com/paper/2203.11025/) (2022)
- [HAMLET: Graph Transformer Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2402.03541/) (2024)
- [Learning from Linear Algebra: A Graph Neural Network Approach to Preconditioner Design for Conjugate Gradient Solvers](https://mlsanity.com/paper/2405.15557/) (2024)
- [A multi-agentic framework for real-time, autonomous freeform metasurface design](https://mlsanity.com/paper/2503.20479/) (2025)
- [Freeform Diffractive Metagrating Design Based on Generative Adversarial Networks](https://mlsanity.com/paper/1811.12436/) (2018)
- [Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layers](https://mlsanity.com/paper/2306.17486/) (2023)
- [Towards General Neural Surrogate Solvers with Specialized Neural Accelerators](https://mlsanity.com/paper/2405.02351/) (2024)
- [Multi-objective and categorical global optimization of photonic structures based on ResNet generative neural networks](https://mlsanity.com/paper/2007.12551/) (2020)
- [A Domain-adaptive Physics-informed Neural Network for Inverse Problems of Maxwell's Equations in Heterogeneous Media](https://mlsanity.com/paper/2308.06436/) (2023)
- [Mechanics-Informed Autoencoder Enables Automated Detection and Localization of Unforeseen Structural Damage](https://mlsanity.com/paper/2402.15492/) (2024)
- [A Generalized Schwarz-type Non-overlapping Domain Decomposition Method using Physics-constrained Neural Networks](https://mlsanity.com/paper/2307.12435/) (2023)

## Built on by (most cited first, 5 of 5)

- [Harnessing Photonics for Machine Intelligence](https://mlsanity.com/paper/2604.10841/) (2026)
- [A Differentiable Ray-Wave Framework for Hybrid Refractive-Diffractive System Modeling and Optimization](https://mlsanity.com/paper/2605.15418/) (2026)
- [Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data](https://mlsanity.com/paper/2605.20514/) (2026)
- [Hierarchical rank-evolving representation for physics-informed neural networks](https://mlsanity.com/paper/2608.09483/) (2026)
- [Inductively Scalable, Single-Step Neural Surrogates for Wave-Scattering Inverse Problems](https://mlsanity.com/paper/2608.17344/) (2026)

## Abstract (arXiv)

Neural networks have promise as surrogate partial differential equation (PDE) solvers, but it remains a challenge to use these concepts to solve problems with high accuracy and scalability. In this work, we show that neural network surrogates can combine with iterative algorithms to accurately solve PDE problems featuring different scales, resolutions, and boundary conditions. We develop a subdomain neural operator model that supports arbitrary Robin-type boundary condition inputs, and we show that it can be utilized as a flexible preconditioner to iteratively solve subdomain problems with bounded accuracy. We further show that our subdomain models can facilitate the construction of global coarse spaces to enable accelerated, large scale PDE problem solving based on iterative multilevel domain decomposition. With two-dimensional Maxwell's equations as a model system, we train a single network to simulate large scale problems with different sizes, resolutions, wavelengths, and dielectric media distribution. We further demonstrate the utility of our platform in performing the accurate inverse design of multi-wavelength nanophotonic devices. Our work presents a promising path to building accurate and scalable multi-physics surrogate solvers for large practical problems.