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PINNs for Solving Unsteady Maxwell's Equations: Convergence Issues and Comparative Assessment with Compact Schemes

T0 review · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Configured PINNs with Fourier features, periodic mappings, and causal training match or exceed traditional finite-difference solvers on 1D and 2D unsteady Maxwell test cases, though NTK analysis shows convergence effort does not track spatial error.

desk verdict Useful ablation and NTK diagnostic for PINNs on unsteady Maxwell, but the headline accuracy claims overreach the reported numbers. read the letter →

arxiv 2504.12144 v1 pith:3SKNKCCH submitted 2025-04-16 physics.comp-ph

Gal G. Shaviner , Hemanth Chandravamsi , Shimon Pisnoy , Ziv Chen , Steven H. Frankel This is my paper · ORCID
classification physics.comp-ph
keywords pinnsconvergenceequationssolvingtrainingcompactframeworklearning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Physics-informed neural networks (PINNs) try to solve equations like Maxwell's by training a network to satisfy the equations at many points, instead of building a mesh. This paper compares PINNs against two standard grid solvers, FDTD and a compact Pade scheme, on three test problems: a 1D Gaussian pulse that forms a sharp wavefront, a 2D pulse in a periodic box, and a 2D pulse hitting a dielectric slab.

The authors find that PINNs need extra ingredients to work well. Adding random Fourier features to handle high frequencies, enforcing periodicity as a hard constraint, and training time slices from early to late (causal training) all help. In the 2D periodic case, the full configuration reaches an L2 error around 4%; without Fourier features, the error jumps to 10% or more. In the dielectric case, enforcing spatial periodicity hurts because the medium is not periodic, and the best configuration omits it.

The paper also uses a Neural Tangent Kernel (NTK) analysis to see where the network learns fastest. It reports that the local convergence rate tracks the time-dependent error, but not the spatial error. That suggests the network does not automatically spend more effort on difficult spatial regions. The main weaknesses are that the headline 1D result has no quantitative error measurement, one prose claim contradicts its own ablation table, and the NTK 'convergence rate' is defined as an average eigenvalue rather than the standard NTK minimum eigenvalue.

Extended reading notes

Core claim

Section 6 states: 'PINNs, when equipped with appropriately chosen enhancements such as RFF and domain-aware priors, can match or exceed traditional solvers for Maxwell's equations in structured and smooth domains.' If true, a properly configured PINN is a viable mesh-free alternative to FDTD and Pade for simple unsteady electromagnetic problems, and the NTK diagnostic (Section 5) predicts where training will not converge efficiently.

Load-bearing premise

Section 4.1 claims the PINN 'accurately captures the field distribution and sharply resolves the discontinuity' for the 1D Gaussian pulse, but no error norm is reported and the network uses smooth tanh activations that cannot represent a true derivative discontinuity. If the oscillation-free PINN solution is actually a smeared approximation, the primary evidence for the 'surpass traditional solvers' claim in the conclusions collapses.

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Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

The central comparison depends on several hand-chosen hyperparameters (loss weights, RFF scale, causality rate, temporal period) and on domain assumptions about applying strong-form Maxwell residuals to discontinuous media and using periodic mappings for non-periodic problems. The paper-level diagnostic C(x,t) is a paper-specific quantity.

free parameters (5)
  • Temporal period P_t = not reported (trainable)
    Used in the periodic coordinate mapping Eq. 16 for the temporal coordinate even though the problems are not time-periodic; the paper states it 'was found to improve convergence' (Sec. 3.1).
  • Loss weights lambda_IC, lambda_BC, lambda_res = 1D: 20, 20, 1; 2D: 50, 50, 1 (Eq. 37)
    Hand-selected weights balancing initial, boundary, and residual losses; no sensitivity analysis.
  • RFF scale sigma = 2
    Standard deviation of the Gaussian entries of the random Fourier matrix B in Eq. 17; chosen without justification.
  • Causality rate epsilon = not reported
    Hyperparameter in Eq. 27 that controls how quickly later time steps enter training; value not stated.
  • Network depth L and width = width 64 or 128; depth not stated
    Architecture choices affect all results but the number of hidden layers is not reported.
assumptions (4)
  • domain assumption Strong form of Maxwell's equations with piecewise constant permittivity is valid at the dielectric interface without explicit interface conditions
    In Sec. 4.3 the residual loss uses epsilon(x) with a jump at x=0.4; no interface loss term enforces continuity conditions, and the strong form is undefined at the discontinuity.
  • ad hoc to paper A non-periodic dielectric problem can be treated with periodic boundary conditions because the pulse is well-contained
    Sec. 4.3: 'we exploit the fact that the pulse remains well-contained within the domain over the simulated time, which enables the use of periodic boundary conditions for practical purposes'.
  • ad hoc to paper Trainable temporal periodicity does not harm accuracy on a non-periodic-in-time problem
    Sec. 3.1 introduces P_t to improve convergence; the paper does not check whether the solution is corrupted outside the training window.
  • standard math NTK theory for infinite-width networks approximates finite-width training dynamics
    Sec. 5 invokes NTK [42] and cites [50,51,52] for finite-width validity; this is a standard but not universally tight assumption.
invented entities (1)
  • Local convergence rate C(x,t) defined as the mean NTK eigenvalue
    purpose: Diagnostic to measure how fast the PINN learns in different space-time regions
    Defined in Eq. 39 as sum of eigenvalues divided by N; the paper asserts it indicates convergence rate without deriving it from NTK gradient-flow theory. It is a paper-specific diagnostic, not an externally validated quantity.

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Pith. "Pith review of PINNs for Solving Unsteady Maxwell's Equations: Convergence Issues and Comparative Assessment with Compact Schemes." pith.science (2026). https://pith.science/paper/3SKNKCCH

@misc{pith2026250412144,
  author       = {Pith},
  title        = {Pith review of: PINNs for Solving Unsteady Maxwell's Equations: Convergence Issues and Comparative Assessment with Compact Schemes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SKNKCCH}},
  note         = {Machine review of arXiv:2504.12144}
}
read the original abstract

Physics-Informed Neural Networks (PINNs) have recently emerged as a promising alternative for solving partial differential equations, offering a mesh-free framework that incorporates physical laws directly into the learning process. In this study, we explore the application of PINNs for solving unsteady Maxwell's equations and compare their performance with two established numerical methods: the Finite-Difference Time-Domain (FDTD) method and a compact Pade scheme with filtering. Three benchmark problems are considered, ranging from 1D free-space wave propagation to 2D Gaussian pulses in periodic and dielectric media. We assess the effectiveness of convergence-enhancing strategies for PINNs, including random Fourier features, spatio-temporal periodicity, and temporal causality training. An ablation study highlights that architectural choices must align with the underlying physics. Additionally, we employ a Neural Tangent Kernel framework to examine the spatio-temporal convergence behavior of PINNs. Results show that convergence rates correlate with error over time but not in space, revealing a limitation in how training dynamics allocate learning effort. Overall, this study demonstrates that PINNs, when properly configured, can match or surpass traditional solvers in accuracy and flexibility, though challenges remain in addressing spatial inhomogeneity and adapting training to localized complexity.

Figures

Figures reproduced from arXiv: 2504.12144 by the authors.

Figure 1
Figure 1. Schematic of PINN architectures for solving Maxwell’s equations in 1D (left) and 2D (right). [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Solution for the electric field Ex obtained using (a) FDTD, (b) Pade (4th order), and (c) PINNs at t = 0.8 for a decaying point valued electric pulse located at z = 0. Noticeable oscillations can be seen with FDTD and Pade, particularly near the sharp gradients, while PINNs exhibits an oscillation free and accurate solution. The blue curve indicates the reference solution computed using 1000 spatial points with FDTD… view at source ↗
Figure 3
Figure 3. Total equation loss versus epoch (left), equation loss versus time after 100,000 epochs (middle), and [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: further illustrates the impact of these components on spatial field accuracy. Sub-figures (a)–(e) show qualitative comparisons between PINN-predicted fields (top rows) and their corresponding absolute errors (bottom rows), sorted from best to worst based on ablation co…
Figure 5
Figure 5. Figure 5: Total equation loss versus epoch (left), equation loss versus time after 100,000 epochs (middle), and [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Electric field Ez distribution (top row) and corresponding absolute errors (bottom row) at t = 1.5 for the 2-D pulse interacting with a dielectric medium. Subplots (a)-(e) shows results from different PINN configurations, each compared against a high-resolution Pade re…
Figure 7
Figure 7. Figure 7: Temporal convergence rate C(t) for different PINN configurations is shown before the (a) first epoch and (b) last epoch. The convergence rate is compared at different epochs for PINN configurations with and without temporally causal training. A close agreement is obser…
Figure 8
Figure 8. Figure 8: Instantaneous Ez field (row 1), L2 error (row 2), auto-differentiated time derivative ∂Ez ∂t (row 3), and NTK-based convergence rate (row 4) at different time instants for the pulse-in-a-box case (Sec. 4.2). Notably, the L2 error aligns with regions of high ∂Ez ∂t , in…

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