Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data
For scientific machine learning, this work demonstrates that embedding governing equations into the network architecture can drastically improve accuracy and training speed over soft-constraint methods.
FLASH-MAX is a shallow neural network that exactly satisfies Maxwell's equations by construction, achieving sub-1% relative error from ~1K sparse observations in seconds and single-digit errors from only 100 observations, while maintaining zero PDE residual.
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.