Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data

2026-05-19

AI Summary

This work addresses the challenge of efficiently reconstructing dynamic electromagnetic fields that satisfy Maxwell’s equations from sparse point observations by proposing FLASH-MAX, a shallow neural network architecture in which each hidden neuron directly corresponds to an analytical solution of Maxwell’s equations. This design constructively embeds physical laws into the hypothesis space, achieving zero partial differential equation (PDE) residual without requiring explicit physics-based terms in the loss function. The method thus simultaneously ensures universal approximation capability and strict physical consistency. Experimental results demonstrate that FLASH-MAX achieves relative validation errors below 1% using only approximately 1,000 sparse three-dimensional observation points, with reconstruction completed in seconds. Even when the number of samples is reduced to just 100, the error remains in the single-digit percentage range, significantly outperforming existing approaches.

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.

Problem

• Maxwell's equations
• sparse data
• scientific machine learning
• electromagnetic field reconstruction
• PDE-constrained learning

Innovation

• exact-by-construction
• Maxwell's equations
• scientific machine learning
• universal approximation
• sparse data reconstruction