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Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data
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Proposes FLASH-MAX, a shallow tanh network whose hidden units are exact homogeneous Maxwell solutions (light-cone weight tying plus Noetherian multipliers), trained on sparse pointwise data to reach sub-5% error in seconds with zero PDE residual.
Problem
Reconstructing homogeneous electromagnetic fields from sparse pointwise observations while exactly satisfying Maxwell's equations, where PINNs enforce the PDE only approximately and are slow to converge.
Approach
A shallow network whose neurons are exact Maxwell plane-wave-type solutions: frequency vectors are constrained to the light cone z0^2 = z1^2+z2^2+z3^2 via weight sharing, and vector outputs are multiplied by Noetherian multipliers computed with Macaulay2, so the PDE holds symbolically by construction.
Key results
- FLASH-MAX reaches <5% relative validation error with zero PDE residual in about 8.1 s (Plane Waves, BC) versus 304 s for the in-house PINN and 346 s for S-EPGP.
- With 100 training points, FLASH-MAX attains minimum relative L2 error of 2.6% on Plane Waves and 7.4% on Radial Waves in the IC setup.
- On CPU, FLASH-MAX reaches about 1-2% error on several cases within 10 minutes of training (e.g., Plane Waves IC 1.2%).
Surprising findings
- The in-house PINN fails badly on Radial Waves with BC (93.9% error), and S-EPGP does not converge there.
- Replacing tanh with cosine activation reduced time to 5% error from 488 s-range to 13.5 s in the width ablation.
Limitations
- Restricted to homogeneous, source-free Maxwell equations with constant material parameters; currents, heterogeneous media and interfaces need new parametrizations.
- Initial and boundary observations are enforced by data fitting, not symbolically, so accuracy depends on data and network size.
- Baselines (PINN, QPINN, Evo-PINN) were adapted or in-house, not fully matched reproductions.
Experimental setup
Datasets: synthetic Maxwell solutions (Plane Waves, Radial Waves, Hopf Fibration, Random Solution)
Benchmarks: Plane Waves Radial Waves Hopf Fibration Random Solution
Metrics: relative L2 error PDE residual error compute time to 5% error
Baselines: PINN S-EPGP FEM QPINN Evo-PINN
Builds on: PINN Ehrenpreis-Palamodov theorem S-EPGP Fourier-feature networks
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