Neural Modal Decomposition: Architectural Priors from Observables

Juho Park; Kaushik Sengupta

physics.comp-phSep 13, 2026

Neural Modal Decomposition: Architectural Priors from Observables

Authors: Juho Park, Kaushik Sengupta

Organizations: Princeton University

Abstract

Many engineering building blocks behave as multi-port linear time-invariant systems. RF cavities, photonic devices, and superconducting quantum chips, despite their different underlying physics, all share a common mathematical structure for their port-level response. Each entry of the response matrix is a sum of contributions from a small number of intrinsic resonant modes, the pole-residue form. A model capable of predicting such responses for arbitrary geometries and arbitrary port configurations, while simultaneously extracting the underlying eigenmode structure, would therefore establish a foundational design principle spanning all these domains. We propose a neural framework that learns this modal decomposition end-to-end, supervised only by system-level observables and without supervising the modal parameters themselves. The architecture decomposes into a port-independent pole predictor and two port-dependent coupling predictors whose outputs are combined entry-wise, separating intrinsic from port-dependent features. This factorization yields a single trained model that generalizes to port counts unseen during training, dissolving the $\mathcal{O}(N^2)$ scaling barrier of direct regression. Despite no modal supervision, the freely-parameterized poles converge to physically meaningful eigenmodes, verified by cross-validation against the AAA rational approximation algorithm. We instantiate the framework in radio-frequency electromagnetic surrogate modeling. A model trained only on 2-port data accurately predicts $N$ -port responses unseen during training.

Figures & tables

Figure 1: A universal architectural prior across physical domains. Multi-port LTI systems in RF/microwave engineering, photonic devices, and superconducting quantum chips all admit the same pole-residue structure. Each response entry is a sum over intrinsic modes with port-independent poles $p_{k}$ and port-dependent couplings ( $l_{k,i}$ , $r_{k,j}$ in microwave network theory, $\kappa_{k,i}^{\text{out}},\kappa_{k,j}^{\text{in}}$ in temporal coupled-mode theory, analogous quantities in black-box quantization). The same algebraic prior underlies all three.

Figure 2: Direct regression versus Neural Modal Decomposition. (a) Direct regression on $S$ -parameters, the standard RF approach, requires a separately collected dataset and a separately trained model for each port count $N$ , and offers no path to port configurations unseen at training time. (b) NMD transforms the $S$ training data to the impedance $Z$ (which has the port-invariance property), trains a single model on a fixed port count to predict poles $p_{k}$ and per-port couplings $l_{k,i},r_{k,j}$ , and reconstructs $Z_{ij}(\omega)$ for any port subset by submatrix selection. The output is converted back to $S$ -parameters via the bilinear transformation for community-standard reporting.

Figure 3: Dataset, port arrangements, and architecture. (a) Cavity stack-up. A patterned $18\times 18$ binary metal layer (4 $\mu$ m thick) is embedded between two dielectrics ( $\varepsilon_{\text{eff}}=3.9$ ) over a ground plane, with 2-port $S$ -parameters sampled at 36 frequencies from 30 to 100 GHz. (b) Two ports (red) on the top/bottom edges carry signal and two AC shorts (blue) on the left/right edges are present in all cavities. Cross-edge and same-edge port arrangements are both included in training, with port pixel locations randomized. (c) NMD pipeline. CavityNet encodes the geometry into a cavity latent, PoleNet maps the latent to $K$ complex poles $p_{k}$ , and AmpNet (left/right) is queried per port to produce $l_{k,i}$ and $r_{k,j}$ . The entry-wise synthesis $Z_{ij}(\omega)=\sum_{k}l_{k,i}r_{k,j}/(j\omega-p_{k})$ is converted to $S$ and supervised.

Figure 4: Network discovers physical poles without supervision. Six representative test geometries (the cavity pattern is shown inset in each panel). For each, the top panel shows the Frobenius norm $\|Z(\omega)\|_{F}$ with ground truth (solid) versus prediction (dashed), and the training band shaded green. The bottom panel shows the top-5 dominant poles in the complex frequency plane, with network poles (red) overlapping AAA-extracted reference poles (black) despite no pole supervision. Agreement holds inside and outside the training band, indicating that the network has recovered the physical eigenmodes.

Figure 5: Zero-shot port-count extension. A single model trained only on 2-port data is evaluated on 3-, 4-, and 5-port configurations not seen at any port count during training. Each row corresponds to a port count (2p, 3p, 4p, 5p). The leftmost column shows the cavity with port locations (red, numbered) and AC shorts (blue). Remaining columns show representative $S$ -parameter magnitudes versus frequency, with predictions (red, dashed) closely matching ground truth (black, solid).

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Training dynamics over 200 epochs. Train and validation $\mathcal{L}_{S}$ (S-parameter MAE), $\mathcal{L}_{Z,\log}$ (log-impedance error), and $\mathcal{L}_{\text{gauge}}$ . Train and validation $\mathcal{L}_{S}$ track closely with no overfitting. $\mathcal{L}_{\text{gauge}}$ saturates near $3\times 10^{-4}$ , indicating near-complete gauge fixing.

Component	Parameter	Value
CavityNet	token dimension $D_{t}$	384
	number of tokens $n$	$4\times 5=20$
	pattern noise $\sigma_{\text{pat}}$ / token dropout $p_{\text{tok}}$	0.05 / 0.10
PoleNet	number of poles $K$	32
	layers / heads / FFN dim	2 / 4 / 512
Left-AmpNet	row embedding dim $d_{\text{row}}$	64

Appendix

Table 1: Hyperparameters for the neural modal decomposition model. The two AmpNet heads share the same architecture but have independent weights, producing left coupling

l_{k,i}

and right coupling

r_{k,j}

respectively. The soft reciprocity term

\lambda_{\text{sym}}

is disabled in the reported configuration so that the framework remains applicable to non-reciprocal systems.

Port	Test set size	mean	median	std	p25	p75	p95
2p	26,430	0.0281	0.0221	0.0267	0.0152	0.0328	0.0612
3p	1,000	0.0405	0.0258	0.0627	0.0184	0.0362	0.1075
4p	1,000	0.0433	0.0250	0.0602	0.0184	0.0376	0.1794
5p	1,000	0.0445	0.0245	0.0570	0.0183	0.0363	0.1925

Appendix

Table 2: Distribution of per-sample

S

-parameter MAE across port counts. The mean reported in the main text is included for reference. Across all held-out port counts the median sits substantially below the mean, indicating that the generalization error is concentrated in a small upper tail rather than spread uniformly across the test set.

Figure 7: Zero-shot port-count extension, Additional example 1.

Figure 8: Zero-shot port-count extension, Additional example 2.

Model	$K$	$\lambda_{g}$	2p	3p	4p	5p
K=8	8	0.5	0.0752	0.0995	0.0944	0.0887
K=16	16	0.5	0.0306	0.0533	0.0533	0.0569
K=32 with gauge fix (main result)	32	0.5	0.0281	0.0405	0.0433	0.0445
K=32 without gauge fix	32	0.0	0.0333	0.0439	0.0438	0.0453
K=64	64	0.5	0.0279	0.0423	0.0431	0.0458

Appendix

Table 3: Ablation on pole count

K

and gauge-fixing weight

\lambda_{g}

. S-parameter MAE on the 2-port in-distribution test set and on held-out 3-, 4-, and 5-port test sets (zero-shot port-count extension). The default configuration used throughout the main paper is K=32 with gauge fix.

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