Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data

TL;DR

FLASH-MAX is a neural network architecture that exactly encodes Maxwell's equations, enabling rapid and precise electromagnetic field predictions from sparse data with minimal training time.

Contribution

The paper introduces FLASH-MAX, a neural network that incorporates Maxwell's equations directly into its structure, achieving fast, accurate predictions from sparse data.

Findings

Achieves sub-1% validation error with about 1K observations

Maintains low errors with only 100 observations in 3D space

Training time is seconds due to symbolic satisfaction of PDEs

Abstract

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.