用神经网络+迭代法高效求解多尺度麦克斯韦方程，支持复杂边界条件。
Accurate and scalable deep Maxwell solvers using multilevel iterative methods
- 构建子域神经算子模型，可处理任意罗宾型边界条件。
- 单个网络训练后可模拟不同尺寸、波长和介质分布的大规模问题。
- 适用于多波长纳米光子器件逆向设计，适合大规模物理仿真场景。
神经网络有望作为偏微分方程（PDE）的代理求解器，但实现高精度与可扩展性仍具挑战。本文表明，神经网络代理可结合迭代算法，精确求解具有不同尺度、分辨率和边界条件的PDE问题。我们开发了一种支持任意罗宾型边界条件输入的子域神经算子模型，并证明其可作为灵活预条件器，迭代求解子域问题并保持有限误差。进一步，该子域模型可用于构建全局粗空间，基于多级域分解实现加速的大规模PDE求解。以二维麦克斯韦方程为模型系统，我们训练单一网络，即可模拟不同尺寸、分辨率、波长及介电介质分布的大规模问题。此外，我们展示了该平台在多波长纳米光子器件高精度逆向设计中的实用性。本工作为构建准确且可扩展的多物理场代理求解器提供了可行路径。
原文摘要 · Abstract (English)
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.
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