Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data
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Computer Science > Machine Learning
Title:Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data
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.
| Comments: | 31 pages, 8 figures |
| Subjects: | Machine Learning (cs.LG) |
| ACM classes: | I.6.3; I.6.5; J.2 |
| Cite as: | arXiv:2605.20514 [cs.LG] |
| (or arXiv:2605.20514v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.20514
arXiv-issued DOI via DataCite (pending registration)
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