Time-Domain Data-Driven Loewner Reduced-Order Models for Electromagnetic Simulation

• Full-wave simulation captures multiscale electromagnetic behavior, but fine meshes and long high-Q transients make repeated design-loop runs costly.The combined spatial and temporal demands can require hundreds of thousands or millions of degrees of freedom and tens or hundreds of thousands of time steps.Source: §I · p. 1
• Conventional projection-based model reduction often requires access to discretized system matrices or operations on the underlying operators.That requirement can be difficult to meet with commercial solvers that expose only input-output responses.Source: §I · p. 1
• State-space system identification instead asks whether input-output data alone can yield a realization that reproduces observed responses and predicts unseen behavior.The solver is treated as a black box rather than accessed through its internal operators.Source: §I · pp. 1–2
• The Loewner framework constructs descriptor realizations algebraically from sampled transfer-function data, with singular-value analysis guiding lower-order selection.Prior work adapted the framework to time-domain input-output trajectories by inferring frequency-response data from transient simulations.Source: §I · p. 2
• The paper formalizes time-domain Loewner identification and prediction in an implementation-level algorithm, then validates it on four electromagnetic structures.The tested FOM sizes span about 136,000 to 479,000 degrees of freedom, and prediction windows reach up to 50 times the training-window duration.Source: §I · p. 2

A. Frequency-Domain Data and Loewner Matrices

• Retained, sufficiently excited frequency-response samples are split into equal, disjoint left and right interpolation sets to define the Loewner and shifted Loewner matrices.The interpolation points are distinct points in the complex plane.Source: A. Frequency-Domain Data and Loewner… · p. 2

B. Reduced-Order Realization

• The reduced descriptor model is formed by projecting the Loewner pencil onto dominant left and right singular-vector subspaces.For a single-input, single-output system, the reduced state has order r; the descriptor matrix need not be invertible if the system is well posed.Source: B. Reduced-Order Realization · pp. 2–3
• The reduced order r is the smallest order whose cumulative singular-value energy exceeds the prescribed threshold η.The singular values are computed from the horizontally and vertically stacked Loewner matrices.Source: B. Reduced-Order Realization · p. 3
• The threshold η is selected heuristically and differs across the validation cases.Cases B and C use η = 1 − 10^-7, case A uses η = 1 − 10^-14, and case D uses η = 1 − 10^-15.Source: B. Reduced-Order Realization · p. 3

• The predicted output spectrum is obtained by applying the reduced transfer function to the prediction-interval input spectrum, followed by an inverse FFT.Without zero-padding, the inverse FFT treats the prediction window as periodic, so a response that has not decayed wraps to the beginning and can create a small pre-excitation ripple.Source: C. Time-Domain Prediction and… · p. 3
• The prediction interval can exceed the training window, with its frequency grid set by observation duration and time resolution set by the sampling interval.The input spectrum and reduced transfer function must be evaluated on a consistent frequency grid.Source: C. Time-Domain Prediction and… · p. 3
• The realization is not guaranteed stable, but reported predictions evaluate its transfer function on the imaginary axis rather than time-stepping the descriptor system.Direct time-stepping or use outside the fitted band would require removing or reflecting right-half-plane poles; passivity enforcement is also left for future work.Source: C. Time-Domain Prediction and… · p. 3
• Identification costs O(n^2) for Loewner assembly and O(n^3) for dense SVDs, so it may outweigh direct simulation for short runs or small FOMs.The benefit is expected for large problems, long predictions, or repeated simulations with different excitations, where identification cost can be amortized.Source: C. Time-Domain Prediction and… · p. 3
• Prediction error is measured separately over the training and extrapolation intervals using normalized root-mean-square error.The denominator uses the norm of the complete reference signal so extrapolation error is not inflated after the response decays.Source: C. Time-Domain Prediction and… · p. 3

• Frequency bins outside the sufficiently excited band are discarded because input-output data may not identify the response there.A fit band may extend beyond the nominal excitation band when spectral tails retain enough energy to resolve structural resonances.Source: D. Numerical Implementation Details · p. 3
• When the input mean is sufficiently far from zero, the ratio of output mean to input mean estimates the DC gain.The estimate may be unreliable if the training window is too short, signals have not reached relevant steady state, or the mean input is near zero; the implementation substitutes it at zero frequency.Source: D. Numerical Implementation Details · pp. 3–4
• Alternating retained samples between left and right interpolation sets on the uniform FFT grid prevents coincident interpolation points.Consequently, the divided-difference denominators do not vanish and the entries require no special treatment.Source: D. Numerical Implementation Details · p. 4

E. Identification and Prediction Algorithm

• The algorithm requires full-order input-output snapshots, a sampling interval, training duration, frequency-selection criteria, and an SVD truncation threshold.Source: E. Identification and Prediction… · p. 4
• It extracts training-window signals, computes their FFTs, selects excited frequency bins, estimates transfer-function samples, and partitions them into left and right interpolation data.Source: E. Identification and Prediction… · p. 4
• The procedure assembles Loewner and shifted Loewner matrices, uses SVDs and a truncation criterion to select the reduced order, then constructs the reduced realization.Source: E. Identification and Prediction… · p. 4
• The reduced transfer function is evaluated on the prediction frequency grid, multiplied by the predicted input spectrum, and inverse-transformed to obtain the predicted response.Source: E. Identification and Prediction… · p. 4

III. VALIDATION

• Four microwave structures are validated against full-order reference data generated with in-house FEM-TD/ROM-TD solvers, using Gaussian-pulse excitation and short-window training followed by full-window extrapolation.Source: §III · p. 4
• The validation figures show each structure’s FOM discretization mesh, FOM-versus-ROM scattering parameters, and corresponding time-domain output for each port.Source: §III · p. 4, Figure 1 · p. 4
• Across cases, the same identification code is used; only the FOM snapshots and three user-set parameters—the fit band, training-window length, and SVD energy threshold—change.Source: §III · p. 4

A. Case A: QM-SIW Filter

• 136,352 to 15 FOM order and 48,200 to 95.6 ms per simulated nanosecond: the ROM trained on 5.5 ns predicts the QM-SIW response over 300 ns.Full-window execution falls from about 4 h to under 5 min; the training window is less than 2% of the full window.Source: A. Case A: QM-SIW Filter · p. 4, Figure 2 · p. 5
• The three reflection minima are reproduced, but the predicted minima are shallower and the third shifts upward by about 60 MHz; predicted return loss remains better than 23 dB from 3.7–4.1 GHz.The first two minima are within about 30 MHz of the reference; predicted depths are −31.5, −32.5, and −38 dB versus reference depths of −40, −43, and −43 dB.Source: A. Case A: QM-SIW Filter · pp. 4–5
• S12 stays within 1 dB of the reference from 3.5–4.25 GHz, then diverges above about 4.5 GHz, reaching −28 dB versus −33.5 dB at 4.6 GHz.The case has a moderate-Q, well-separated pole structure and uses a 5.5 ns training window to predict 300 ns, about a 54-fold extrapolation.Source: A. Case A: QM-SIW Filter · p. 5

B. Case B: 3-D FSS

• The FOM order falls from 285,974 to 12 for reflection and 11 for transmission, while per-nanosecond cost falls from 51,800 to 8.35 ms, with accurate extrapolation to 40 ns.The ROMs use a 10.5 ns training window.Source: B. Case B: 3-D FSS · p. 5, Figure 7 · p. 6, Figure 8 · p. 6
• The reflection ROM captures seven expected return-loss zeros individually, consistent with the coupled-resonator behavior of the dual-polarization unit cell.The zeros correspond to H-shaped resonators, rather than an averaged envelope.Source: B. Case B: 3-D FSS · pp. 5–6, Figure 5 · p. 6
• The main disagreement is at the fitted band’s lower edge: predicted transmission differs by up to about 1.3 dB between 12.0 and 12.3 GHz.The passage attributes this to the excitation spectrum falling away there and notes that a longer training window could improve lower-frequency passband accuracy.Source: B. Case B: 3-D FSS · p. 6

C. Case C: Broadband Combline Filter

• A 13th-order ROM reduces the 371,220th-order FOM, with per-nanosecond cost falling from 366,600 to 4.75 ms, and reproduces the eight-ripple passband and transitions into both stopbands almost exactly between roughly 7 and 17 GHz.The 2.1 ns training window supports validation over the widest relative bandwidth of the four cases.Source: C. Case C: Broadband Combline Filter · pp. 6–7
• The eight reflection minima are tracked individually, and predicted and reference curves agree within about 3 dB over the central 8–15 GHz region.Source: C. Case C: Broadband Combline Filter · p. 7
• Outside roughly 7–17 GHz, predictions diverge: below 7 GHz no transfer-function samples constrain the model, while above 18 GHz numerical poles yield a non-passive response.The paper advises against using the ROM outside its roughly 7–17 GHz reliable range.Source: C. Case C: Broadband Combline Filter · p. 7

D. Case D: Diplexer

• The diplexer ROM reduces FOM order from 478,778 to 18 and per-nanosecond cost from 102,900 to 1.04 ms.The simulation uses 2,400 snapshots at δt = 0.05 ns.Source: D. Case D: Diplexer · p. 7
• Within passbands, reflection minima can differ by several decibels: near 1.696 GHz the ROM predicts −33.5 dB versus −38.3 dB, and near 1.775 GHz −38.5 dB versus −29.7 dB.The shallow minimum near 1.82 GHz is not resolved; narrow ripples relative to frequency-sample spacing contribute to the mismatch.Source: D. Case D: Diplexer · p. 8
• The 18th-order ROM correctly separates both channel paths and matches passband edges and the shared-junction transition for the closely spaced Tx/Rx channels.The reported 120 ns training window yields predictions over the full 300 ns window.Source: D. Case D: Diplexer · p. 7, Figure 14 · p. 8
• Out-of-band isolation is not reliable: reference transmissions fall to −44 to −69 dB, while predicted S21 levels off at −37 to −52 dB and S31 at −28 to −34 dB.The authors caution against relying on this ROM for deep stopband rejection or channel-isolation figures.Source: D. Case D: Diplexer · p. 8
• Across the four cases, ROM order rises from 11 to 18 while per-nanosecond speed-up ranges from 504× to nearly 105×; accuracy and cost depend on different factors.The authors associate error with usable information per frequency sample and pole excitation, while extrapolation ratio appears to affect savings more than error.Source: D. Case D: Diplexer · pp. 8–9

E. Sensitivity to Training-Window Length

• The shortest window in each case's sweep consistently gives the lowest ROM order and least accurate fit, often with a spurious resonance or unbounded band-edge excursion.Table II summarizes the swept window lengths and resulting orders.Source: E. Sensitivity to Training-Window Length · p. 9, Table II · p. 10
• For case A, the 4 ns model merges three reflection zeros, the 5.5 ns model resolves them 8–10 dB too shallow, and the 8 ns model tracks minima within a few dB.All three models place the nulls near 3.8–4.0 GHz at essentially the correct frequencies; longer windows improve their depth and separation.Source: E. Sensitivity to Training-Window Length · p. 10, Figure 18 · p. 10
• Closely spaced poles become distinguishable only after the transient beat develops; the training window sets frequency resolution for separating nearby resonances.At fit-band edges below roughly 1 GHz and above 5 GHz, decaying input spectrum makes samples unreliable and identified orders place different spurious poles.Source: E. Sensitivity to Training-Window Length · p. 10
• Repeating identification for two or three candidate window lengths is an inexpensive, data-driven diagnostic: check whether the predicted transfer function stabilizes.Steps 2–7 cost little relative to the full-order simulation.Source: E. Sensitivity to Training-Window Length · p. 10

IV. CONCLUSIONS

• Across four cases, ROM order was 3–4 orders lower than FOM order, with high-fidelity response prediction beyond training (up to 54 times longer for case A and 29 times longer for case C).Per-nanosecond evaluation cost fell by up to five orders of magnitude, and end-to-end run time fell from hours to minutes in two of four cases; fidelity was highest in the passband and degraded below roughly 25–40 dB of dynamic range.Source: §IV · p. 10
• Future work includes improving numerical stability and transient behavior, predicting outputs for unseen inputs, and extending the approach to nonlinear systems.Larger-scale EM problems are also identified as a promising next target.Source: §IV · p. 10
• Algorithm 1 requires only time-domain excitation and response snapshots, allowing use with existing workflows without modifying the underlying solver.This is intended as a low-effort first step for groups without access to, or unwilling to modify, full-order simulation matrices.Source: §IV · p. 10
• All four validation cases use synthetic, essentially noisefree FOM transients rather than measurements, which would add sensor noise and calibration error.The paper notes that Algorithm 1’s masking safeguards would need tuning for measured data.Source: §IV · pp. 10–11
• Training-window sweeps were conducted case by case, leaving a quantitative stopping rule for choosing t_train as future work.The paper also identifies passivity enforcement as future work to improve robustness.Source: §IV · p. 11
• Results across four structures with different topologies, bandwidths, and FOM sizes suggest training-window sensitivity is associated with Loewner identification rather than a particular solver or geometry.The method is presented in implementation-level form to support direct reproducibility from time-domain excitation and response snapshots.Source: §IV · p. 11