用神经网络精确求解麦克斯韦方程，仅需少量数据秒级完成
Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data
- 每个隐藏层节点对应一个麦克斯韦方程的精确解，结构上天然满足物理规律
- 1000个采样点下相对误差低于1%，100个点时误差仍保持个位数
- 适合需要快速高精度模拟电磁场的科研与工程场景
我们提出FLASH-MAX，一种浅层、构造性精确的神经网络架构，用于从稀疏点观测中预测均匀电磁场。每个隐藏神经元代表麦克斯韦方程的一个独立精确解，使网络在构建时即符号满足控制方程，可从稀疏数据端到端训练，耗时仅数秒。我们证明了该精确模型类在任意域上仍具有普遍逼近能力。FLASH-MAX在约1000个稀疏点观测下实现低于1%的相对验证误差，且始终保持零偏微分方程残差；即使仅使用3D空间中100个观测点，误差仍维持在个位数水平。结果表明，将控制结构从损失函数转移到假设类中，能显著提升科学机器学习中精度与优化速度的权衡。
原文摘要 · Abstract (English)
We introduce FLASH-MAX, a shallow, exact-by-construction neural network architecture for predicting homogeneous electromagnetic fields from sparse pointwise observations. Each hidden neuron represents a separate exact solution to Maxwell's equations, so that the network satisfies the governing equations symbolically by construction and can be trained end-to-end from sparse data within seconds. We prove a universal approximation result showing that this exact model class remains universal on arbitrary domains. FLASH-MAX reaches sub-1% relative validation error from about 1K sparse pointwise observations in seconds, all while maintaining a zero PDE residual, and keeps single-digit errors even for only 100 observations sampled from 3D space. These results suggest that moving governing structure from the loss into the hypothesis class can dramatically improve the trade-off between precision and optimization speed in scientific machine learning.
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