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REVIEW 2 major objections 4 minor 45 references

Second-order AAA algorithms for structured data-driven modeling

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Three greedy algorithms fit second-order mechanical models straight from frequency data.

desk verdict A genuinely useful second-order AAA extension with shipped code and an honest heuristic analysis, but Eq (47) has a missing w_j in the SO-AAA Jacobian that has to be fixed. read the letter →

arxiv 2506.02241 v1 pith:LEUHUTZ5 submitted 2025-06-02 math.NA cs.LGcs.NAcs.SYeess.SYmath.DSmath.OC

Michael S. Ackermann , Ion Victor Gosea , Serkan Gugercin , Steffen W. R. Werner This is my paper
classification math.NAcs.LGcs.NAcs.SYeess.SYmath.DSmath.OC MSC 41A2065D1593B1593C0593C80
keywords second-ordersystemsadaptiveAntoulas-Andersonalgorithmstructuredbarycentricformfrequency-domaindata-drivenmodelingrationalapproximationvariableprojectionLoewnermatrixmodelorderreduction
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 extends the AAA greedy rational-approximation algorithm to build second-order dynamical systems directly from frequency-response measurements. It presents three variants that trade speed against accuracy: LSO-AAA keeps the quasi-support points fixed for a fast linear solve, SO-AAA optimizes them with variable projection, and NSO-AAA optimizes the full nonlinear residual. The central claim is that these methods adaptively choose the model order to meet a tolerance while preserving the $M\ddot{x} + D\dot{x} + Kx = bu$ structure, so the learned model can be interpreted as a mass-damper-stiffness system.

What carries the argument

The key object is the second-order structured barycentric form (14), where each pair $(\lambda_j, \sigma_j)$ generates a quadratic pole structure and the transfer function automatically has the form $c^T (s^2 I + s D + K)^{-1} b$. The greedy selection is the AAA maximum-error sample choice; the least-squares fit is done through the Loewner-like matrix $L_{\mathrm{SO}}$ with entries $(g_i - h_j)/((\mu_i - \lambda_j)(\mu_i - \sigma_j))$ and, for NSO-AAA, replaced by the current model value $H_{\mathrm{SO}}(\mu_i)$. The quasi-support points $\sigma_j$ are initialized far in the left half-plane by $\sigma_j = c - i\,\mathrm{Im}(\lambda_j)$, and Theorem 2 shows that as $|\mathrm{Re}(\sigma_j)| \to \infty$ the second-order model converges to the unstructured AAA model, giving an upper bound heuristic for the structured methods' error.

What would settle it

Rerun the gyroscope experiment with $\sigma_j$ moved into the right half-plane or exactly on the imaginary axis, and observe whether SO-AAA's and NSO-AAA's frequency errors collapse; the same experiment with the acoustic cavity would show whether the accuracy gain comes from the structure or from the initial pole placement.

Watch

Extended reading notes

Core claim

The paper's central claim is that the structured second-order barycentric form $H_{\mathrm{SO}}(s) = \frac{\sum_{j=1}^k h_j w_j}{\prod (s-\lambda_j)(s-\sigma_j)}$ over the same sum in the denominator can be driven by the AAA greedy selection: pick the next support point as the current model's worst-data point, then fit the barycentric weights (and quasi-support points) by least squares. The authors introduce three algorithms: LSO-AAA leaves the $\sigma_j$ fixed so each iteration is one linear solve, SO-AAA uses variable projection on the separable residual with Jacobians derived from the Loewner-like matrix, and NSO-AAA solves the fully nonlinear residual with Wirtinger calculus. In the numerical examples, SO-AAA and NSO-AAA typically give much smaller errors than unstructured AAA at the same model order, and in the acoustic-cavity example they even beat AAA at twice the order.

Load-bearing premise

The whole accuracy story depends on the quasi-support point initialization $\sigma_j = c - i\,\mathrm{Im}(\lambda_j)$ with a single hand-picked far-left constant $c$; LSO-AAA never moves these points, and the two nonlinear variants merely start from them and may settle in poor local minima.

Editorial extensions

If this is right

  • Practitioners who only have frequency-domain data of a mechanical, acoustic, or electrical structure can obtain a model that retains mass-damper-stiffness form, ready for physical interpretation and time-domain simulation.
  • The adaptive greedy order selection gives a principled stopping criterion: keep adding support points until a user-set tolerance on the weighted data error is met.
  • The theory predicts that the structured methods cannot be much worse than unstructured AAA at the same order, because the $\sigma_j$ initialization reproduces the AAA model when the quasi-support points are far enough from the data.
  • The realification section supplies real-matrix state-space realizations, so the learned models can pass directly to standard time integrators and control design tools.
  • For applications needing very fast model updates, LSO-AAA performs a single linear least-squares solve per order, with accuracy essentially matching classical AAA at equal order.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable next step is to apply the $\sigma_j$ initialization rule to other structure-preserving barycentric forms (damped, gyroscopic, or second-order with multiple inputs), since the theorem's mechanism only uses the partial-fraction expansion and the far-field limit of the quasi-support points.
  • The observed discrepancy between the separable cost and the true nonlinear error suggests a diagnostic: track both the optimization objective and a validation error, and stop when the two decouple, which practitioners could automate in a wrapper around the method.
  • The gyroscope example hints that NSO-AAA may lose to SO-AAA when transfer-function magnitudes span many orders of magnitude; a plausible extension is to rescale or frequency-warp the data before the fully nonlinear fit to keep gradient computations well conditioned.
  • Formal guarantees for the greedy selection would likely require an equioscillation or Kolmogorov-type condition on the second-order model class; the paper's heuristics give the floor and ceiling but not a certificate for the greedy step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops three variants of the AAA algorithm for fitting second-order state-space models to frequency response data. The methods share a greedy interpolation step that selects support points from the data, and differ in how they determine the barycentric weights and quasi-support points: LSO-AAA fixes the quasi-support points and solves a linear least-squares problem, SO-AAA uses a separable (linearized) residual with variable projection, and NSO-AAA solves the fully nonlinear residual. The authors also provide representability results (Lemmas 2-3, Theorem 1), an error heuristic driven by the behavior of the second-order barycentric form when the quasi-support points move far into the left half-plane (Theorem 2), and a construction of real state-space realizations. Three numerical examples (sandwich beam, butterfly gyroscope, acoustic cavity) compare the proposed methods against unstructured AAA and against AAA with twice the order.

Significance. If the technical issues are resolved, this is a valuable contribution to structured data-driven modeling. The central idea of imposing second-order differential structure through a barycentric form is natural and useful, and the three algorithmic variants offer a practical speed-accuracy trade-off. The paper is generally well written, and the provision of open source code and data (Zenodo) is a genuine strength. The representability lemmas and the structural comparison to unstructured AAA provide useful insight. The numerical experiments cover relevant benchmarks and show that the nonlinear variants often outperform unstructured AAA at the same order, which supports the motivating claim. The main weakness is that the derivation of the SO-AAA Jacobian contains an error that affects the algorithm's correctness, and this must be fixed and validated before the results can be fully trusted.

major comments (2)
  1. [Section 4.2, Eq. (47)] Equation (47) is incorrect: it omits the barycentric weight w_j in the Jacobian of the separable residual with respect to the quasi-support point σ_j. From Eq. (43c), the residual entry is r_i = η_i (Σ_j w_j (h_j − g_i)/((μ_i − λ_j)(μ_i − σ_j)) − g_i) = −η_i (Σ_j w_j [L_SO]_{i,j} + g_i). Differentiating with respect to σ_j gives ∂r_i/∂σ_j = −η_i w_j [L_SO]_{i,j}/(μ_i − σ_j). The factor w_j is missing in Eq. (47). This is not a harmless typo because the weights change at every iteration and affect the descent direction and fixed points of the Gauss-Newton/VarPro iteration. The analogous Jacobian for NSO-AAA in Eq. (55) contains the w_j factor, and the realification version of the SO-AAA Jacobian in Eq. (98) again omits it. The authors should confirm whether the released code implements Eq. (47) as printed or uses a corrected Jacobian; if the printed formula was used, the SO-AAA results in Section 7 must be recomputed with a correct Jacobian.
  2. [Section 4.2, text after Eq. (45)] The text says 'We observed that this selection causes the objective function (45) to decrease monotonically as the order of approximation k increases' and the value of the offset c in Eq. (46) is chosen as −10^5 in the experiments. This is an empirical observation, not a proven property, and the convergence of SO-AAA and the accuracy of LSO-AAA (which never moves σ_j) depend on this heuristic. The theoretical analysis in Section 5.2 only provides an asymptotic statement as min_j |Re(σ_j)| → ∞. The paper should state clearly that the monotonicity claim and the specific choice of c are heuristics without a rigorous guarantee, and ideally add a brief experiment on the sensitivity to c.
minor comments (4)
  1. [Section 6, Eq. (89b) and Eq. (101b)] In Eq. (89b) and Eq. (101b), the entry for row 2i, column 2j is written as [L]_{2i−1,2j−1}, which is the same index as the first entry of the previous line. The intended entry is presumably [L]_{2i,2j} = −Re(α_{i,j} − β_{i,j} in Eq. (89b) and the analogous correction in Eq. (101b).
  2. [Section 2.3, first paragraph] The abstract and Section 2 contain the typo 'varibale' instead of 'variable' in 'varibale projection method'.
  3. [Section 5.2, Theorem 2] The theorem states the limit in the L∞ norm but does not specify the domain (the imaginary axis is clear from the proof but should be stated in the theorem). Also, the condition 'if L† has constant rank in some open neighborhood' is imprecise about the neighborhood (of what? presumably the limit point at infinity); a more careful statement would help.
  4. [Section 6, text before Eq. (96)] There is a typo 'matirx' instead of 'matrix' in the sentence introducing the notation L_SO = L_SO(P_SO, M).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained given the published structured barycentric form [16], and the heuristic claims are explicitly empirical.

full rationale

Walking the derivation chain, the central objects are the second-order barycentric form (14) and realization (16), imported from the authors' earlier published paper [16]. That is a prior peer-reviewed derivation, not an unverified self-citation invoked to forbid alternatives, and the present paper derives its own algorithmic consequences: the greedy selection, the linearized residual (43), the VarPro formulation (45), the Jacobians (47) and (54)-(56), the realification in Section 6, and the switching results in Lemmas 2-3 and Theorem 1. The sigma initialization heuristic (46) is explicitly empirical and is not used to prove the main accuracy claims; Theorem 2 is a direct limit statement showing LSO-AAA approaches AAA as sigma moves far into the left half-plane, and it is derived rather than assumed. The upper-bound heuristics in Section 5.2 are presented as heuristics, and the paper itself flags in the conclusions that simplified cost functions can misalign with true errors. Numerical validation is against external benchmarks (sandwich beam, butterfly gyroscope, acoustic cavity) with standard AAA and AAA2 baselines and released code and data. The only substantive issue noticed is a likely missing w_j factor in the published Jacobian (47) when compared with (55); that is a correctness or typographical concern, not circular reasoning, so it does not affect the circularity score.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The algorithms introduce quasi-support points as model parameters (fitted in SO/NSO, fixed in LSO); the initialization offset c and NSO initial weight -1 are hand-chosen heuristics. The central representation is borrowed from the authors' prior paper [16], and the monotone-decrease claim is an assumption.

free parameters (3)
  • Quasi-support points sigma_j = Optimized per iteration; initialized as c - i Im(lambda_k)
    In SO-AAA and NSO-AAA, sigma_j are fitted to data by nonlinear least squares; in LSO-AAA they are fixed at initialization. These are the extra parameters that give the second-order structure.
  • Initialization offset c = -10^5 in experiments
    Hand-chosen large negative real part for sigma_j. Theorem 2 assumes Re(sigma_j) -> -inf; finite c is a tuning constant and LSO-AAA's accuracy depends on it.
  • Initial barycentric weight w_k in NSO-AAA = -1
    Heuristic initialization for the newly added weight; the paper states it yields monotone decrease of the objective. No derivation is given.
assumptions (5)
  • domain assumption The second-order barycentric form (14) can represent transfer functions of second-order systems (1) under mild assumptions.
    Invoked in Section 2.2 and used for all three algorithms; taken from [16] without re-derivation.
  • standard math Any irreducible strictly proper rational function of degree k can be represented by the interpolatory barycentric form (6).
    Lemma 1, proved in the paper using the Loewner matrix; standard in rational interpolation.
  • standard math The Loewner matrix L in (11) has full rank for data from a degree-k rational function.
    Invoked in Lemma 1, cited from [3].
  • standard math The pseudoinverse is continuous where the matrix has constant rank.
    Used in Theorem 2 to pass to the limit bL_SO^dagger -> L^dagger; cited from Stewart [37].
  • ad hoc to paper Initializing sigma_k far in the left half-plane makes the objective (45) decrease monotonically in k.
    Stated as an empirical observation in Section 4.2 after eq. (46); assumed in the design of the algorithms and in the heuristics of Section 5.2.

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Cite this review

Pith. "Pith review of Second-order AAA algorithms for structured data-driven modeling." pith.science (2026). https://pith.science/paper/LEUHUTZ5

@misc{pith2026250602241,
  author       = {Pith},
  title        = {Pith review of: Second-order AAA algorithms for structured data-driven modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LEUHUTZ5}},
  note         = {Machine review of arXiv:2506.02241}
}
read the original abstract

The data-driven modeling of dynamical systems has become an essential tool for the construction of accurate computational models from real-world data. In this process, the inherent differential structures underlying the considered physical phenomena are often neglected making the reinterpretation of the learned models in a physically meaningful sense very challenging. In this work, we present three data-driven modeling approaches for the construction of dynamical systems with second-order differential structure directly from frequency domain data. Based on the second-order structured barycentric form, we extend the well-known Adaptive Antoulas-Anderson algorithm to the case of second-order systems. Depending on the available computational resources, we propose variations of the proposed method that prioritize either higher computation speed or greater modeling accuracy, and we present a theoretical analysis for the expected accuracy and performance of the proposed methods. Three numerical examples demonstrate the effectiveness of our new structured approaches in comparison to classical unstructured data-driven modeling.

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