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M\"obius Reparametrization of Multiport-Network Models of PIN-Diode-Programmable Metasurfaces for Accurate Low-Order Neumann Approximations

Philipp del Hougne

arXiv:2609.37557v1eess.SPphysics.app-ph

TL;DR

MNT models capture mutual coupling in programmable metasurfaces through a costly configuration-dependent matrix inversion, limiting gradient-based optimization. The paper derives a closed-form Möbius reparametrization for low-order Neumann approximations and evaluates it on a strongly coupled fabricated DMA. Its second-order approximation closely approaches full-model classification accuracy while reducing evaluation time and saved-tensor memory.

  • Problem

    Configuration-dependent MNT matrix inversion is costly in gradient-based optimization, and the accuracy of finite-order Neumann approximations depends on the chosen parametrization.

    Source: §I · p. 1, §III · p. 3

  • Method

    The paper derives a closed-form Möbius reparametrization for 1-bit elements and uses it in truncated forward and adjoint Neumann evaluations for gradient-based optimization.

    Source: §I · p. 1, §II · p. 2, §III · p. 3

  • Results

    At K = 2, forward and VJP accuracies reach 21.68 and 21.70 dB; with M = 8, test accuracy is 94.97% vs. 95.57% for full proxy MNT.

    Source: §IV · p. 4

  • Takeaways & Limitations

    The best achievable zeroth-order model accuracy provides a parametrization-invariant MC-unaware benchmark and MC-strength metric.

    Source: §VI · p. 5

Abstract

from arXiv · show

Accurate models of programmable metasurfaces based on multiport-network theory (MNT) account for mutual coupling (MC) through a configuration-dependent matrix inversion. The latter's high computational cost during gradient-based optimization (GBO) can be alleviated via a finite-order Neumann approximation. We show that the accuracy of this approximation depends strongly on the (tacitly) chosen MNT parametrization. For 1-bit-programmable meta-elements (e.g., meta-elements based on PIN diodes), we identify a closed-form M\"obius reparametrization that depends only on the meta-element's two load states and requires neither training data nor numerical optimization. For an experimentally estimated proxy MNT model of a fabricated 96-element 19-GHz dynamic metasurface antenna with strong MC, a second-order Neumann approximation with our reparametrization achieves forward and control-gradient accuracies of 21.68 and 21.70 dB, respectively. Relative to the full proxy MNT, it reduces the runtime of a combined forward and control-gradient evaluation by a third and the saved-tensor memory by a factor of ten. In a prototypical GBO problem of end-to-end optimization for DMA-based scene classification, it achieves 94.97% test accuracy vs. 95.57% with the full model. We further note that MC-strength metrics and MC-unaware benchmarks should be parametrization-invariant, motivating, for instance, definitions based on the best directly fitted zeroth-order model.

I. INTRODUCTION

MNT captures mutual coupling through a configuration-dependent matrix inversion, but repeated forward and adjoint evaluations make that inversion costly in gradient-based optimization. This work studies whether reparametrization can improve low-order Neumann approximations, including for experimentally calibrated systems with strong coupling.

I. INTRODUCTION

  • Gradient-based optimization repeatedly needs forward and adjoint evaluations, making configuration-dependent MNT matrix inversion costly as the numbers of elements and configurations grow.Coordinate-descent methods can use Woodbury low-rank updates, but GBO iterations generally modify all element states.Source: §I · p. 1
  • Prior truncated-Neumann work simplified MNT-based RIS optimization and parameter estimation, but had not used low-order approximations for both forward and adjoint models in GBO.This gap matters because both evaluations are needed at each GBO iteration.Source: §I · p. 1
  • Distinct proxy MNT parameter sets can reproduce the same end-to-end measurements, so reparametrization can alter internal parameters without changing measurable responses.Such ambiguity is usually operationally irrelevant, but requires care in segmented estimation and has also been used to tighten an ambiguity-sensitive bound.Source: §I · p. 1
  • The paper derives a closed-form Möbius reparametrization for 1-bit elements and evaluates low-order forward and adjoint approximations in end-to-end optimization of a strongly coupled, fabricated DMA.The study also identifies a parametrization-invariant mutual-coupling metric and an MC-unaware benchmark model.Source: §I · pp. 1–2
  • Strong mutual coupling makes the fabricated 96-element PIN-diode DMA a relevant test because without coupling the Neumann series reduces exactly to its zeroth-order term.The paper notes that strong coupling can enhance wave-domain flexibility while complicating accurate modeling and optimization.Source: §I · p. 1
  • Modified Born-series methods address Neumann convergence or low-iteration accuracy in non-reconfigurable scattering systems, but not reparametrizable MNT models of programmable metasurfaces.The paper identifies a lack of fixed-low-order forward and adjoint GBO evaluation investigated with an experimentally calibrated proxy model.Source: §I · p. 2

II. SYSTEM MODEL

The system model represents tunable elements as loads on virtual ports and expresses the end-to-end channel and its control gradient through MNT. Finite-order Neumann approximations avoid matrix inversions, but their accuracy depends on the MNT parametrization.

II. SYSTEM MODEL

  • Each 1-bit-tunable lumped element is modeled as a load terminating a virtual port, and the tunable subsystem’s scattering matrix is diagonal in the element reflection coefficients.The system is partitioned into static and tunable subsystems interconnected through the virtual ports.Source: §II · p. 2
  • The end-to-end channel matrix H maps control states to the channel through blocks D, A, B, and Γ of the static subsystem’s scattering matrix.These blocks correspond to output-input, output-virtual, virtual-input, and virtual-virtual port sets, respectively.Source: §II · p. 2
  • The control gradient follows by differentiating the channel response with respect to each relaxed control variable and substituting the 1-bit load derivative.The adjoint-variable method computes the shared factor once and reuses it across all element gradients.Source: §II · p. 2
  • For the passive, lossy, radiatively open systems considered, the ground-truth round-trip operator has spectral radius below unity, but an end-to-end-fitted proxy parametrization need not preserve this condition.The stated convergence condition applies to both the forward Neumann series and its adjoint.Source: §II · p. 2
  • Truncating the forward and adjoint Neumann series at order K yields recursive evaluations without matrix inversions, with the kth forward term representing k rescattering events.The implicit approximate adjoint uses the order-K forward and adjoint approximations, whereas differentiating the truncated forward model gives an unrolled gradient; they need not coincide.Source: §II · p. 2
  • Möbius reparametrization preserves the exact channel and derivatives but changes finite-order approximations and their truncation errors.The transformation changes the MNT parameters while leaving the exact mapping from controls to channel unchanged.Source: §II · pp. 2–3

III. M ¨OBIUS REPARAMETRIZATION

The exact MNT response and control derivatives are invariant to Möbius reparametrization, but finite-order Neumann approximations are not. The proposed closed-form choice uses the two binary load states to reduce truncation error, alongside numerical benchmarks for order-K approximations.

Motivation

  • Finite-order Neumann forward and adjoint approximations vary with Möbius reparametrization even though the exact response and derivatives do not.The reparametrization is therefore chosen to reduce truncation errors in both quantities.Source: §III · p. 3

Closed Form

  • For lossy passive load states with |α|, |β| < 1, the closed-form choice minimizes q(m) uniformly across binary configurations and is the hyperbolic midpoint of α and β.It maps the two transformed load states to opposites, MmCF(α) = −MmCF(β).Source: §III · p. 3

Numerical Benchmarks

  • The numerical benchmarks define separate forward- and adjoint-optimal Möbius parameters for each approximation order.The adjoint benchmark compares the full-MNT control gradient with an order-K estimate computed using the truncated forward model and implicit adjoint.Source: §III · p. 3
  • The forward and control-gradient benchmark parameters are estimated separately using a coarse search followed by multistart Nelder–Mead optimization.The loss sensitivity is held fixed as the Möbius parameter varies, so the gradient comparison probes the MNT gradient approximation.Source: §III · p. 3
  • The directly fitted zeroth-order affine model is a less restricted K = 0 benchmark because all its parameters are fit freely.Its initialization uses complex least squares with rank-one SVD projection, followed by full-batch gradient-based optimization of normalized response mean-squared error.Source: §III · p. 3

IV. RESULTS

In the calibrated strong-coupling DMA, mCF supports accurate second-order forward and control-gradient approximations with lower runtime and saved-tensor memory. Its end-to-end scene-classification accuracy closely tracks training with the full proxy MNT.

Setup and Sensing Task

  • Evaluation uses a calibrated 96-element, eight-feed 19-GHz DMA with strong cavity coupling and jointly optimizes DMA configurations with a classifier for MNIST scene sensing.The scene reflectivity vector encodes an MNIST image, and final test accuracy uses the full proxy MNT.Source: §IV · pp. 3–4

Truncated-Model Accuracy

  • After reparametrization, the series converges for the all-eβ configuration, with the spectral radius of eΦeΓ between 0.82 and 0.92.The initial proxy parametrization does not guarantee convergence for this configuration.Source: §IV · p. 4
  • The order-K truncated T-model accuracy depends strongly on m, and mCF visually coincides with the numerical forward and gradient optima.The same qualitative pattern holds for y’s forward error and for control-gradient errors associated with both T and y.Source: Figure 2 · p. 4, §IV · p. 4
  • Without reparametrization, convergence is uncertain and K ≤ 2 truncations have essentially no predictive value.At K = 0, mCF accuracy is comparable to independently fitted rank-one models.Source: §IV · pp. 3–4
  • At K = 2, mCF reaches 21.68 dB forward accuracy and 21.70 dB VJP accuracy; evaluation time falls from 14.32 to 9.09 ms and saved-tensor memory from 19.72 to 1.95 MiB versus full MNT.The accuracies are closely matched at this order.Source: §IV · p. 4, Table II · p. 5

Scene-Classification Accuracy

  • 94.97% vs. 95.57% test accuracy at M = 8: mCF with K = 2 closely tracks full-proxy-MNT training in DMA-based scene classification.Final test accuracies are evaluated using the full proxy MNT model.Source: §IV · p. 4, Figure 3 · p. 5

V. PARAMETRIZATION-INVARIANT MC-STRENGTH METRIC AND MC-UNAWARE BENCHMARK MODEL

Setting Γ = 0 does not define a unique MC-unaware benchmark because Möbius reparametrizations preserve end-to-end response while changing Γ. The section instead proposes a directly fitted rank-one affine model whose minimum fitting error is parametrization-invariant.

V. PARAMETRIZATION-INVARIANT MC-STRENGTH METRIC AND MC-UNAWARE BENCHMARK MODEL

  • The Γ = 0 benchmark error depends on MNT parametrization and can become arbitrarily large as I_NM − mΓ approaches a singularity.Möbius reparametrizations preserve the end-to-end response while changing Γ.Source: §V · p. 5
  • The minimum fitting error of a directly fitted rank-one affine model provides a parametrization-invariant MC-unaware benchmark and measures MC-induced nonlinearity.The fitted parameters need not be unique.Source: §V · p. 5
  • 6.86 dB and 5.89 dB are the joint T-and-R fit values of ζR and ζT, respectively, for the strongly coupled 1-bit DMA.The metric was evaluated for binary controls; multibit controls can generalize the model by making g_n a scalar function of the element’s load state.Source: §V · p. 5

VI. CONCLUSION

A closed-form Möbius reparametrization enables accurate low-order Neumann approximations of forward and adjoint MNT evaluations under strong mutual coupling. The second-order approximation reduces runtime and memory while closely approaching full-MNT scene-classification performance.

VI. CONCLUSION

  • 21.68 dB and 21.70 dB are the forward and control-gradient accuracies of the reparametrized second-order approximation for an experimentally calibrated proxy MNT model under strong mutual coupling.The evaluation reduces computational time by about one third and saved-tensor memory by a factor of ten.Source: §VI · p. 5
  • 95.0% versus 95.6% at M = 8 are the scene-classification accuracies of the reparametrized second-order-truncated MNT model and full MNT, respectively.The case study uses end-to-end-optimized sensing.Source: §VI · p. 5
  • The best achievable zeroth-order model accuracy provides a parametrization-invariant MC-unaware benchmark and MC-strength metric.Source: §VI · p. 5
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