mlsanityData to 14 Oct 2026

Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data

arXiv 2605.20514 ↗First posted 19 May 2026cs.LG cs.NA math.NAcited by 0

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Abstract (arXiv)

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.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 26 of its 78 references that are in the map. First posted on arXiv 19 May 2026.

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References · 26

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,238
  3. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  5. 2018On the Spectral Bias of Neural Networkscited by 1,341
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  7. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
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  10. 2017Linearly constrained Gaussian processescited by 42
  11. 2015Modeling and interpolation of the ambient magnetic field by Gaussian processescited by 28
  12. 2018Algorithmic Linearly Constrained Gaussian Processescited by 20
14 more
  1. 2023A neural operator-based surrogate solver for free-form electromagnetic inverse designcited by 19
  2. 2022NeurOLight: A Physics-Agnostic Neural Operator Enabling Parametric Photonic Device Simulationcited by 17
  3. 2022Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficientscited by 17
  4. 2020Linearly Constrained Gaussian Processes with Boundary Conditionscited by 14
  5. 2022Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equationscited by 14
  6. 2025Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunitiescited by 11
  7. 2024Exact Enforcement of Temporal Continuity in Sequential Physics-Informed Neural Networkscited by 9
  8. 2023A Domain-adaptive Physics-informed Neural Network for Inverse Problems of Maxwell's Equations in Heterogeneous Mediacited by 7
  9. 2024PACE: Pacing Operator Learning to Accurate Optical Field Simulation for Complicated Photonic Devicescited by 5
  10. 2025Accurate and scalable deep Maxwell solvers using multilevel iterative methodscited by 5
  11. 2022On boundary conditions parametrized by analytic functionscited by 4
  12. 2023Physics-constrained 3D Convolutional Neural Networks for Electrodynamicscited by 3
  13. 2024Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equationscited by 2
  14. 2025Gaussian Process Regression for Inverse Problems in Linear PDEscited by 2

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