Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data

Dan DeGenaro1, Xin Li2, Obed Amo3, Michael Pokojovy3,4, Sarah Adel Bargal1, Markus Lange-Hegermann5, Bogdan Raiţă2

1Department of Computer Science, Georgetown University   2Department of Mathematics, Georgetown University   3Department of Mathematics and Statistics, Old Dominion University   4School of Data Science, Old Dominion University   5Institute Industrial IT, Department of Computer Science and Automation, OWL University of Applied Sciences and Arts  

NeurIPS 2026 Spotlight

arXiv Code Camera-ready (soon) Supplement (soon)

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.

Results

Both electric and magnetic fields as predicted by FLASH-MAX are nearly indistinguishable from the simulated ground-truth solutions.

Timing

We plot FLASH-MAX's relative error over time against 3 baselines (PINN, S-EPGP, FEM), plotting only points where the residual is reasonable (< 0.01). FLASH-MAX consistently outperforms all techniques by guaranteeing a zero residual while quickly and reliably achieving high accuracy across problems and setups. FEM does not appear as it cannot achieve a small residual this quickly.

Timing plot 1
Only initial conditions enforced.
Timing plot 2
Boundary and initial conditions enforced.

Method

Method overview
From sparse pointwise observations of a single field instance, FLASH-MAX fits a shallow trainable architecture whose hidden neurons are exact Maxwell solutions, so the predicted field solves Maxwell’s system throughout optimization. The resulting model is exact by construction, expressive on arbitrary domains, and yields accurate reconstruction from sparse data. Exactness is ensured by using computer algebra to obtain the hidden layer restrictions on z and the polynomials p(z) which tie the vectorial weights. Top-left: Sparse data sampled only at t=0 (“×” symbols), showing that our model is predictive; data can also be sampled on the boundary of the validation domain (“•” symbols). Bottom: Quiver plot visualization of our fast prediction of an electromagnetic knot (Hopfion) at time t=0.5. The difference to ground truth is indistinguishable.

Citation

@inproceedings{degenaro2026flashmax,
  title     = {{Fast} {Reconstruction} of {Exact} {Maxwell} {Dynamics} from {Sparse} {Data}},
  author    = {{DeGenaro}, {Dan} and {Li}, {Xin} and {Amo}, {Obed} and {Pokojovy}, {Michael} and {Bargal}, {Sarah Adel} and {Lange-Hegermann}, Markus and {Raiţă}, Bogdan},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2026},
  notes     = {Accepted as a spotlight poster presentation.}
}