Accurate and scalable deep Maxwell solvers using multilevel iterative methods

2025-09-03

AI Summary

To address the limited accuracy and scalability of traditional PDE solvers for multiscale problems, variable-resolution discretizations, and complex geometries with heterogeneous material distributions, this paper proposes a multilevel domain decomposition framework based on subdomain neural operators to build a scalable deep Maxwell solver. Our key contributions are: (1) a subdomain neural operator designed to enforce arbitrary Robin boundary conditions, ensuring both generalization capability and physical consistency; (2) integration of this operator as a flexible preconditioner to construct a global coarse space, enabling efficient multilevel iterative solution; and (3) end-to-end joint training for both forward electromagnetic simulation and inverse photonic design tasks. Experiments demonstrate that a single trained model achieves high-accuracy solutions for both forward and inverse nanophotonic problems—spanning multiple wavelengths, resolution levels, and heterogeneous media—while significantly improving computational efficiency and scalability to large-scale problems.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data

Abstract

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.

Problem

• Solving Maxwell's equations with high accuracy and scalability
• Developing neural network surrogates for multi-scale PDE problems
• Enabling inverse design of nanophotonic devices through improved solvers

Innovation

• Neural network surrogates combined with iterative algorithms
• Subdomain neural operator model with arbitrary boundary conditions
• Multilevel domain decomposition for accelerated large-scale solving