Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data

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.