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Physics-Informed Neural Networks for the Time-Domain Maxwell Equations with Split-Field Perfectly Matched Layers

T0 review · 2 major / 0 minor · reviewed 2026-06-25 · grok-4.3

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Pith's one-line read Physics-informed neural networks solve time-domain Maxwell equations in open domains by incorporating split-field PMLs into the loss.

desk verdict Split-field PMLs let the same Maxwell equations go into the PINN loss everywhere, and the method matches references on basic 1D/2D pulses, but the support is only visual and the tests stay elementary. read the letter →

arxiv 2606.25323 v1 pith:4ZW3E4TE submitted 2026-06-24 math-ph math.MPphysics.comp-ph

Xiaodong Liu , Lingquan Li , Gregory Moss , Scott Langdon This is my paper · ORCID
classification math-phmath.MPphysics.comp-ph
keywords physics-informedneuralnetworksMaxwellequationssplit-fieldPMLtime-domainelectromagneticsopen-domainsimulationsGaussianpulsepropagation
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

The paper establishes a PINN formulation for the time-domain Maxwell equations that incorporates split-field perfectly matched layers. This approach uses the same governing equations in both the physical domain and the PML regions, which simplifies loss construction. Validation on one-dimensional and two-dimensional Gaussian pulse problems shows that the resulting solutions agree with analytical and FDTD reference solutions.

What carries the argument

The split-field perfectly matched layer formulation, which absorbs outgoing waves while preserving identical Maxwell equations in physical and PML regions.

What would settle it

A significant deviation between the PINN solution and the FDTD reference on the reported one- or two-dimensional Gaussian pulse problems, especially in the PML region where waves should decay without reflection, would show the formulation fails to deliver accurate results.

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Extended reading notes

Core claim

By embedding the auxiliary equations of the split-field PML directly into the PINN loss function, the same set of equations governs wave propagation throughout the entire domain, enabling accurate open-domain solutions without separate boundary treatments or region-specific modifications.

Load-bearing premise

A standard neural-network optimizer can reliably converge to an accurate solution of the coupled PDE system when the split-field PML auxiliary equations are included in the loss without additional stabilization or weighting adjustments.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 0 minor

Summary. The paper presents a PINN formulation for the time-domain Maxwell equations that incorporates split-field PMLs. A stated advantage is that the same governing equations apply in both the physical and PML regions, simplifying loss construction. The method is validated on 1D and 2D Gaussian pulse problems, with the claim that PINN solutions show good agreement with analytical and FDTD references, demonstrating feasibility for open-domain time-domain EM simulations.

Significance. If the implementation details and quantitative support are supplied, the work would establish the feasibility of a uniform-equation PINN approach for Maxwell problems with split-field PMLs on simple low-dimensional test cases. This could reduce the complexity of loss-term construction relative to other PML formulations, though the demonstration is restricted to short-time Gaussian pulses without claims of generality or superiority over FDTD.

major comments (2)
  1. [Abstract and numerical results] Abstract and validation sections: the claim of 'good agreement' is asserted without any reported network architecture (layers, width, activations), loss-term weights, optimizer hyperparameters, training schedule, or convergence diagnostics. These elements are required to evaluate whether the optimizer reliably solves the coupled system that includes the split-field auxiliary equations.
  2. [Numerical experiments] Validation sections: support for the central feasibility claim rests solely on visual field plots; no quantitative error measures (relative L2 norms, pointwise maxima, or integrated residuals) or comparison tables against the FDTD reference are supplied, preventing assessment of accuracy beyond qualitative inspection.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments correctly identify gaps in the presentation of implementation details and quantitative validation. We will revise the manuscript to address both points fully.

read point-by-point responses

  1. Referee: [Abstract and numerical results] Abstract and validation sections: the claim of 'good agreement' is asserted without any reported network architecture (layers, width, activations), loss-term weights, optimizer hyperparameters, training schedule, or convergence diagnostics. These elements are required to evaluate whether the optimizer reliably solves the coupled system that includes the split-field auxiliary equations.

    Authors: We agree that these implementation details are essential for reproducibility and for assessing whether the optimizer successfully handles the coupled split-field system. In the revised manuscript we will add a new subsection (or appendix) that reports the network architecture (depth, width, activation functions), the relative weights assigned to each term in the composite loss, the optimizer and its hyperparameters, the training schedule (including any adaptive weighting or curriculum strategies), and convergence diagnostics such as loss histories and residual norms. These additions will directly address the concern about reliable solution of the auxiliary equations. revision: yes

  2. Referee: [Numerical experiments] Validation sections: support for the central feasibility claim rests solely on visual field plots; no quantitative error measures (relative L2 norms, pointwise maxima, or integrated residuals) or comparison tables against the FDTD reference are supplied, preventing assessment of accuracy beyond qualitative inspection.

    Authors: We accept that visual agreement alone is insufficient for a rigorous feasibility demonstration. The revised manuscript will include quantitative error metrics: relative L2 norms of the electric and magnetic field components over the computational domain at selected time instants, maximum pointwise errors, and integrated residual norms of the governing equations. We will also add a table that directly compares these quantities against the FDTD reference solutions for both the 1-D and 2-D test cases. This will allow readers to evaluate accuracy beyond qualitative inspection. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper applies a standard PINN loss construction (residuals of Maxwell's equations plus initial/boundary conditions) to the split-field PML formulation taken from prior literature. Validation is performed by direct numerical comparison against independent analytical solutions and FDTD reference data on 1D/2D Gaussian-pulse test cases. No step equates a derived quantity to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames an empirical pattern as a new derivation. The central feasibility result therefore rests on external benchmarks rather than internal self-reference.

Assumptions & free parameters 1 free parameters · 2 assumptions · 0 invented entities

The approach rests on standard Maxwell equations and the established split-field PML formulation; the neural-network component introduces many implicit hyperparameters whose values are not reported.

free parameters (1)
  • Neural network weights and architecture hyperparameters
    Weights are fitted during training; specific layer counts, activation functions, and collocation point densities are not stated in the abstract.
assumptions (2)
  • domain assumption Maxwell's equations govern the electromagnetic fields in both physical and PML regions
    Invoked as the governing physics throughout the domain.
  • domain assumption Split-field PML auxiliary equations correctly absorb outgoing waves without reflection
    Taken from prior FDTD literature and assumed to transfer directly to the PINN loss.

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Cite this review

Pith. "Pith review of Physics-Informed Neural Networks for the Time-Domain Maxwell Equations with Split-Field Perfectly Matched Layers." pith.science (2026). https://pith.science/paper/4ZW3E4TE

@misc{pith2026260625323,
  author       = {Pith},
  title        = {Pith review of: Physics-Informed Neural Networks for the Time-Domain Maxwell Equations with Split-Field Perfectly Matched Layers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ZW3E4TE}},
  note         = {Machine review of arXiv:2606.25323}
}
read the original abstract

Physics informed neural networks (PINNs) incorporate Maxwell's equations, initial conditions, boundary conditions, and measurement data directly into the learning process, transforming the solution of partial differential equations into a constrained optimization problem. As such, PINNs are attracting increased attention in computational electromagnetics as an alternative to traditional time-domain solvers. This paper presents a PINN formulation for the time-domain Maxwell's equations incorporating split-field perfectly matched layers (PMLs). A key advantage of the split-field PML formulation is that the same governing equations can be applied in both the physical and PML regions, simplifying the PINN formulation and loss construction. The proposed approach is validated using one-dimensional and two-dimensional Gaussian pulse problems with PML. The PINN solutions show good agreement with analytical and finite-difference time-domain (FDTD) reference solutions, demonstrating the feasibility of combining PINNs with split-field PMLs for open-domain time-domain electromagnetic simulations.

Figures

Figures reproduced from arXiv: 2606.25323 by the authors.

Figure 1
Figure 1. (a) Electric Ez and magnetic Hy fields. (b) Comparison between PINNs and the exact solution at t = 0.8. Unless otherwise specified, Latin hypercube sampling (LHS) is adopted for data generation, and the hyperbolic tangent (tanh) function is used as the network activation. The resulting optimization problem is solved using a two-stage training strategy: the Adam optimizer [17] is first employed for stochastic gradien… view at source ↗
Figure 2
Figure 2. (a) Electric Ez and magnetic Hy fields. (The white line represent the interface at x = 0.5) (b) Comparison between PINNs and the exact solution at t = 1.2. The third case utilizes a PML region. The domain is (x, t) ∈ [0, 1] × [0, 1.5]. For x < 0.6, the medium is vacuum with εr = 1.0; for x > 0.6, the region is modeled as PML with εr = 1.0. Here, Np = 15000, N0 = 1000, Nb = 1000 and Nm = 150. The network consists of … view at source ↗
Figure 3
Figure 3. (a) Electric Ez and magnetic Hy fields. (b) Comparison between PINNs and FDTD at t = 0.8. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: (a) Electric field Ez at t = 0.2. (b) The solution comparison between PINNs and FDTD at x = 0.5 and t = 0.2. 4 Conclusions and Future Work This paper explored split-field PML in the context of PINNs. The results show good agreement with reference solutions, demonstrati…
Figure 5
Figure 5. Figure 5: Electric field Ez at (a) t =0.4; (b) t = 0.56. [2] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media”, IEEE Trans. Antennas Propag., vol. 14, no. 3, pp. 302—307, May 1966. 1 [3] J. Jin, The Finite Element…

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Reference graph

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