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arxiv: 2509.06130 · v2 · submitted 2025-09-07 · ⚛️ physics.optics

A general framework for knowledge integration in machine learning for electromagnetic scattering using quasinormal modes

Viktor A. Lilja , Albin J. Sv\"ardsby , Timo Gahlmann , Philippe Tassin This is my paper

Pith reviewed 2026-05-18 17:59 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords quasinormal modeselectromagnetic scatteringphysics-informed neural networksphotonic crystalsmetasurfacesscattering matrixenergy conservationcausality
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The pith

Neural networks constrained by quasinormal modes learn resonant scattering structures while obeying energy conservation and causality.

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

This paper presents a framework that embeds the quasinormal mode expansion of the scattering matrix into neural network models for electromagnetic scattering. The networks are trained to learn the resonant contributions rather than fitting arbitrary functions to the spectra. Because the expansion naturally respects energy conservation and causality, the resulting models inherit these properties without needing explicit penalties during training. Demonstrations on photonic-crystal slabs and free-form metasurfaces show that the approach achieves good accuracy with much less training data than standard networks. The method is general enough to apply to many optical devices and allows manual addition of extra constraints like symmetries when available.

Core claim

By basing the neural network architecture on the quasinormal mode expansion of the scattering matrix, the models learn the underlying resonant structure of the scattering spectrum, are guaranteed to obey energy conservation and causality, and exhibit significantly improved data efficiency for photonic-crystal slabs and all-dielectric free-form metasurfaces.

What carries the argument

The quasinormal mode expansion of the scattering matrix, which decomposes the scattering response into a sum of resonant terms that automatically satisfy physical constraints such as causality and energy conservation.

If this is right

  • Models require substantially fewer training examples to achieve accurate predictions for photonic structures.
  • Additional physical constraints such as losslessness or geometric symmetries can be imposed directly on the network.
  • The framework extends to a broad class of electromagnetic devices due to the generality of the quasinormal mode formalism.
  • Predictions remain physically valid even for inputs outside the training distribution in terms of conservation laws.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar modal decompositions could be used to inform neural networks in other wave scattering problems, such as acoustics or quantum mechanics.
  • This integration of prior physics knowledge may reduce the need for large datasets in inverse design tasks for nanophotonics.
  • Future work could test the framework's performance on structures with many overlapping resonances where truncation errors might appear.

Load-bearing premise

The quasinormal mode expansion must provide a sufficiently accurate and complete representation of the scattering matrix for the devices of interest, with negligible truncation error.

What would settle it

Train the network on a device with known significant truncation error in the quasinormal mode expansion and check whether the predicted scattering spectra violate energy conservation or causality when compared to full electromagnetic simulations.

Figures

Figures reproduced from arXiv: 2509.06130 by Albin J. Sv\"ardsby, Philippe Tassin, Timo Gahlmann, Viktor A. Lilja.

Figure 1
Figure 1. Figure 1: FIG. 1. Relation between the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Overview of the QNM-Net architecture. White boxes [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Applying the QNM-Net to PhC slabs. (a) Illustration [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. QNM-Net applied to free-form metasurface. (a) Il [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Effect of design noise on model predictions. Left [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Effect of spectrum noise on model predictions. Left [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
read the original abstract

Neural networks have been demonstrated to be able to accelerate the modeling and inverse design of optical and electromagnetic devices by serving as fast surrogates for electromagnetic solvers. Nevertheless, such neural networks can be unreliable and normally require extreme amounts of data to train. Here it is shown that these limitations can be alleviated by constraining neural-network models using prior knowledge about the governing physics. We propose a universal physics-informed neural network framework for electromagnetic scattering based on the quasinormal mode expansion of the scattering matrix. The neural networks learn the resonant structure underlying the scattering spectrum, are guaranteed to obey energy conservation and causality, and are shown to have significantly improved data efficiency for photonic-crystal slabs and all-dielectric free-form metasurfaces. Furthermore, the framework allows additional problem-specific constraints, such as losslessness, symmetries, and number of modes, to be imposed manually when they are available. The method can be applied to a wide range of optical and electromagnetic devices owing to the generality of the quasinormal mode formalism.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a universal physics-informed neural network framework for electromagnetic scattering based on the quasinormal mode (QNM) expansion of the scattering matrix. Neural networks learn resonant parameters while the modal structure enforces energy conservation and causality by construction; additional constraints such as losslessness or symmetries can be imposed manually. The approach is demonstrated on photonic-crystal slabs and all-dielectric free-form metasurfaces, with claims of significantly improved data efficiency.

Significance. If the central claims are substantiated, the work would provide a general, extensible route for embedding established modal physics into machine-learning surrogates for optics. This could improve model reliability and reduce data requirements for inverse design and fast modeling tasks across a range of electromagnetic devices.

major comments (2)
  1. [Methods (QNM expansion and truncation)] The guarantees of exact energy conservation and causality rest on the assumption that a finite QNM expansion sufficiently represents the scattering matrix for the target devices. For all-dielectric free-form metasurfaces the manuscript does not report quantitative reconstruction error (e.g., norm of the residual between the truncated and full scattering matrix) as a function of the manually chosen number of retained modes, leaving the exactness of the enforced constraints unverified.
  2. [Results (data-efficiency experiments)] The reported data-efficiency gains for free-form metasurfaces are presented without an accompanying ablation that isolates the contribution of the QNM constraints from possible under-resolution of the modal basis. Explicit comparison of validation error versus number of retained modes is needed to confirm that the observed improvements are not an artifact of an incomplete expansion.
minor comments (2)
  1. [Abstract and Introduction] The abstract and introduction would benefit from a concise statement of the precise form of the truncated QNM expansion used for the scattering matrix, including how the residual continuum is treated.
  2. [Figures] Figure captions should explicitly state the number of QNMs retained in each plotted comparison so that readers can assess truncation level directly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We address each major comment below and will incorporate the suggested analyses in the revised manuscript.

read point-by-point responses

  1. Referee: [Methods (QNM expansion and truncation)] The guarantees of exact energy conservation and causality rest on the assumption that a finite QNM expansion sufficiently represents the scattering matrix for the target devices. For all-dielectric free-form metasurfaces the manuscript does not report quantitative reconstruction error (e.g., norm of the residual between the truncated and full scattering matrix) as a function of the manually chosen number of retained modes, leaving the exactness of the enforced constraints unverified.

    Authors: We agree that quantitative reconstruction errors are needed to verify the modal truncation. In the revised manuscript we will add the norm of the residual between the truncated and full scattering matrix as a function of the number of retained modes for the all-dielectric free-form metasurfaces, confirming that the chosen basis is sufficient for the enforced constraints. revision: yes

  2. Referee: [Results (data-efficiency experiments)] The reported data-efficiency gains for free-form metasurfaces are presented without an accompanying ablation that isolates the contribution of the QNM constraints from possible under-resolution of the modal basis. Explicit comparison of validation error versus number of retained modes is needed to confirm that the observed improvements are not an artifact of an incomplete expansion.

    Authors: We acknowledge the value of an explicit ablation. The revised manuscript will include validation error versus number of retained modes for the free-form metasurfaces, isolating the contribution of the QNM constraints from possible basis truncation effects and confirming that the reported data-efficiency gains are not an artifact of an incomplete expansion. revision: yes

Circularity Check

0 steps flagged

No circularity: framework applies established external QNM formalism to NN parameter learning

full rationale

The derivation chain relies on the quasinormal-mode expansion of the scattering matrix, which is an established result from prior literature in electromagnetic theory rather than a result derived or fitted within this paper. The neural network learns resonant parameters inside that pre-existing modal structure; the guarantees of energy conservation and causality follow directly from the analytic properties of the QNM expansion itself. No self-citation is load-bearing for the core constraints, no fitted input is relabeled as a prediction, and no ansatz or uniqueness claim is smuggled in via the authors' own prior work. The paper is therefore self-contained against external benchmarks and receives a zero circularity score.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that quasinormal modes form a usable basis for the scattering matrix and on the modeling choice of how many modes to retain; no new physical entities are postulated.

free parameters (1)
  • number of retained modes
    The framework permits manual specification of the number of modes; this choice affects the approximation quality and is problem-dependent rather than derived from first principles.
axioms (1)
  • domain assumption Quasinormal modes provide a complete or sufficiently accurate expansion basis for the electromagnetic scattering matrix.
    This premise is invoked to justify representing the scattering spectrum through the neural network's learning of modal parameters.

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    Based on the theoretical foundations of the QNM ex- pansion, we know that ˜ ω m correspond exactly to eigen- frequencies Maxwell’s equations ( 2)

    are ex- plainable in terms of the network-predicted physics pa- rameters, which for the QNM-Net are C(ω ), ˜ω m, dm, and τn. Based on the theoretical foundations of the QNM ex- pansion, we know that ˜ ω m correspond exactly to eigen- frequencies Maxwell’s equations ( 2). Thus, the accuracy of the learned physics can be verified by comparing the network-pre...

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