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Koopman operator based identification of nonlinear networks

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that a two-step Koopman-operator method recovers both the interaction graph and the local dynamics of heterogeneous nonlinear networks from input-state snapshots, with data cost set by sparsity rather than dictionary size.

desk verdict A solid two-step Koopman network identification method with strong numerics, but the edge-detection guarantee is void for the experiments shown. read the letter →

arxiv 2412.18492 v1 pith:LDQUXMO7 submitted 2024-12-24 eess.SY cs.SY

Ramachandran Anantharaman , Alexandre Mauroy This is my paper · ORCID
classification eess.SYcs.SY MSC 93B30
keywords KoopmanoperatornetworkidentificationBooleanreconstructionnonlinearsystemslifteddynamicssystemsparsenetworksdynamicmodedecomposition
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

This paper develops a two-step, data-driven method for reconstructing both the wiring and the local dynamics of a continuous-time nonlinear network whose nodes may have different internal dynamics, external inputs, and pairwise coupling functions. The first step recovers, for every node, which other nodes and inputs affect it, using a Koopman-operator approximation of the vector field and per-node functions whose orthogonal projections produce an edge weight. The second step restricts estimation to the discovered neighbors and recovers the coefficients of the local dynamics and couplings from a lifted linear model. If the method works as claimed, a sparse network can be fully identified with far fewer measurement samples than the number of dictionary functions needed to describe the dynamics, which is why the approach scales to networks with over a thousand nodes.

What carries the argument

The machinery is the dual Koopman method: approximate the Koopman operator and its generator on the finite space of samples rather than on a chosen dictionary, use Gaussian radial-basis test functions selected to minimize the vector-field approximation error, and read off the vector-field values at the sample points. Edge detection then uses per-node subspaces $F_{x,k}=\mathrm{span}\{\phi_{kl}\}$ and $F_{u,k}=\mathrm{span}\{\psi_{kl}\}$; the zero-mean conditions (25)-(26) make these subspaces mutually orthogonal, so the projection weight $\Lambda_{ik}$ of node $i$'s vector field onto node $k$'s subspace is nonzero exactly when node $k$ influences node $i$ (Theorem 1). Once neighborhoods are known, the local identification step fits lifted matrices $A_i, E_i, B_i$ and converts them back to the unknown coefficients through the continuous-time formulas (43)-(44) and the extraction rule (47).

What would settle it

Take a two-node network with no true edge and node functions {$x_k, x_k^2$} on $[-1,1]$; since $x_k^2$ has nonzero mean, the orthogonality behind Theorem 1 fails, so a noise-free run that still yields $\Lambda_{ik}$ above threshold for the non-neighbor would show that the zero-mean condition is doing the theoretical work and must be enforced in practice.

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Extended reading notes

Core claim

The central claim is that the Boolean interaction graph and the full local dynamics can both be recovered from input-state snapshots without solving a regression over the entire network dictionary. Theorem 1 guarantees that weights $\Lambda_{ik}$ and $\Delta_{ik}$, computed from orthogonal projections of each node's vector-field component onto the spans of node functions, vanish exactly when the corresponding coupling or input term is absent, provided the node functions integrate to zero and the true terms are not orthogonal to those spans. Under that guarantee, a sparse regression on the estimated vector-field samples yields per-node neighbor sets; a second, local Koopman lift then estimates the parameters of $F_i$, $H_{ik}$, and $G_{ik}$. The paper also produces a modular linear lifted model of the whole network, assembled blockwise from the local fits.

Load-bearing premise

The edge-detection guarantee relies on the node functions integrating to zero over the state and input domains so that different nodes' subspaces are orthogonal, and the parameter estimates rely on the chosen local dictionaries being nearly invariant under the Koopman generator; the numerical examples use monomials such as $x_i^2$ and $u_i^2$ that do not integrate to zero, so the no-false-positive guarantee is not operative in those demonstrations.

Editorial extensions

If this is right

  • Boolean network reconstruction and local dynamics identification can be performed from the same dataset, so no separate topological probing experiment is needed.
  • Under network sparsity, the number of samples needed scales with the number of neighbors actually present rather than with the total dictionary size, making large sparse networks tractable.
  • Heterogeneous nodes are handled explicitly: each node gets its own local dictionary, so internal dynamics, coupling functions, and input effects may all differ across nodes.
  • A by-product linear lifted model of the whole network, assembled blockwise from local fits, can be used for analysis and control design.
  • Sampling time $T_s$ directly controls accuracy because the continuous-time parameter formulas come from a zero-order-hold approximation of the lifted dynamics.

Reading between the lines

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

  • A testable extension is to center and orthogonalize node functions empirically against the sampling distribution before computing projections, which could preserve the no-false-positive property for polynomial dictionaries like the monomials used in the numerical demonstrations.
  • Because Theorem 1's converse also requires the true coupling to be non-orthogonal to the node-function subspace, a dictionary that misses a mode of the coupling will misclassify that edge; a diagnostic would be to inspect the residual norm after each projection.
  • The modular lifted model opens a route to distributed prediction and control, since each node's block is constructed independently from local neighbors once the graph is known.
  • A natural stress test is to run the two-step method on partially observed networks, where hidden nodes would likely be absorbed into the recovered local dynamics and could create spurious edges; the method's guarantees appear to require full-state measurement.
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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

3 major / 5 minor

Summary. The manuscript develops a two-step identification method for continuous-time nonlinear networked dynamics with heterogeneous local dynamics, external inputs, and node-dependent coupling functions. In the first step, the authors extend the dual Koopman method to non-autonomous systems and use it to estimate the vector field, then regress it on 'node functions' to determine each node's neighbor set and input set (Boolean topology reconstruction). In the second step, they build a local finite-dimensional Koopman lifted dynamics for each node from the estimated neighbors and recover the coefficients of local and coupling dictionary functions. The main theoretical result, Theorem 1, states that under zero-mean conditions on the node functions, the projection weights Lambda_ik are nonzero exactly when there is an edge k to i. Numerical experiments on Erdos-Renyi graphs, non-polynomial vector fields, and Hindmarsh-Rose neuronal networks support scalability (up to 1500 nodes) and accuracy, and the paper also constructs a modular lifted representation of the network dynamics.

Significance. If the stated guarantees held for the demonstrated configurations, this would be a valuable contribution to nonlinear network identification: it relaxes the homogeneity assumptions of earlier Koopman-based network identification, provides a data-efficiency argument for sparse networks, and yields a modular lifted model that can be reused for control or analysis. The paper includes a broad set of numerical experiments and comparisons with the dual method of [18]. However, the gap between the assumptions of Theorem 1 and the node functions used in Sections 5.1-5.3, together with the unstated invariance assumption in the local identification step, means that the current manuscript does not yet rigorously support the central claim as demonstrated. The core idea is promising and the necessary fixes are identifiable, so the issues are reparable within the scope of the paper.

major comments (3)
  1. [Theorem 1, Section 4.2.2 and Sections 5.1-5.2] The edge-detection guarantee is void for the node functions used in the experiments. Equations (25)-(26) require all node functions to have zero mean, and the proof of Theorem 1 uses this to conclude that F_x,k is orthogonal to F_x,l and to F_u,l. However, in all numerical sections the node functions are {x_i, x_i^2} and {u_i, u_i^2} on [-1,1], and x_i^2 and u_i^2 have mean 2/3, not zero. For a two-node example with no edge 2 to 1 and F_1 = x_1^2, the L2 projection onto span{x_2, x_2^2} has coefficient 5/9 on x_2^2, so the analogue of Lambda_12 is nonzero. Thus the no-false-positive statement does not cover the demonstrated results; the reported AUROC values are obtained through thresholding and Lasso rather than through Theorem 1. The authors should either use zero-mean node functions (for example, shifted or Legendre polynomials) in the experiments, or extend Theorem 1 to general node functions with explicit correction or threshold bounds.
  2. [Section 4.2.2, Eqs. (27)-(33)] The theorem assumes that Lambda_ik is computed from the exact orthogonal projection in (27), but Algorithm 1 computes the coefficients from the Lasso problem (33), which is a different estimator. Even when (25)-(26) hold, a regularized regression does not coincide with the projection, so Theorem 1 does not by itself justify the statement that the estimated Lambda_ik are the projection weights used in the proof. The relation between the Lasso solution and the projection coefficients needs to be quantified, or Theorem 1 needs to be restated for the finite-sample regularized estimator, including the effect of the threshold delta in (34).
  3. [Section 4.3.2, Eqs. (40)-(44) and (47)] The local parameter recovery assumes that the finite-dimensional dictionary span F_i is invariant, or nearly invariant, under the Koopman operator or its generator, so that the discrete lifted matrices in (40) describe an exact finite-dimensional representation and log(Abar_i) in (43) equals the generator matrix restricted to that span. This invariance assumption is not stated. For general dictionaries such as the monomials, sinusoids, and exponentials used in Section 5.2, the span is not invariant, and then the coefficient estimates in (47) are not guaranteed to converge to the true alpha, beta, and gamma. A bound on the effect of the projection error on (47), or a precise approximation assumption, is needed to make the central identification claim rigorous.
minor comments (5)
  1. [Section 3.2, Proposition 1] The proof drops O(T_s^2) terms before taking the limit; a uniform remainder bound over the finite set of test-function choices would be needed to justify the interchange of the limit and the argmin.
  2. [Sections 2.1 and 4.1] The shift operator T^t is called a left shift in Section 2.1 and a right shift in Section 4.1, although both definitions are the same; please make the terminology consistent.
  3. [Section 4.3.1] The symbol Mbar_i is used both for the estimated input set and for the total number of input dictionary functions; please use distinct notations for these two objects.
  4. [Throughout Section 5] The acronym 'RSME' appears in figure captions and Table 1; it should be 'RMSE' for consistency with the definition in Section 5.
  5. [Section 5.2, Eq. (50)] The expressions 'sin(xti)' and 'exti' are typeset in a way that makes it difficult to distinguish the variable x_{t_i} from the exponential e^{x_{t_i}}; please clarify the notation and define all variables in the piecewise dynamics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the topology and local identification results are derived from the dynamics and orthogonality assumptions, not from fitted parameters or self-citation chains.

full rationale

The paper's two-step method is not circular. Theorem 1 derives edge detection from the zero-mean conditions (25)-(26) and the additive structure (21) via orthogonal projection; the conclusion does not presuppose the neighbor sets. The local identification in Section 4.3 estimates parameters by solving a least-squares problem for the lifted dynamics (38)-(47); the estimated parameters are not fed back into the theorem that justifies them. The authors rely on the dual and main methods of [18], but that is a published external method, and the present results are extensions rather than restatements of it. Proposition 1 selects test functions by minimizing a training error, but it is an asymptotic equivalence result, not a prediction claimed against held-out data, and it is not used to produce the headline network identification claims. The numerical experiments use node functions {x_i, x_i^2} and {u_i,u_i^2}, which do not satisfy the zero-mean assumptions of Theorem 1; this is a mismatch between theory and implementation that affects the force of the stated guarantee, but it is a correctness and robustness issue, not a circularity. No central claim reduces by construction to its inputs, so the circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical entities. Its load-bearing ingredients are the user-chosen node functions, thresholds, and regularization parameters, plus standard Koopman approximation assumptions. The mismatch between the zero-mean condition in the theory and the monomials used in experiments is the most consequential item.

free parameters (4)
  • threshold δ = varies in experiments
    User-chosen threshold in (34) for deciding whether Λ_ik or Δ_ik indicates an edge; ROC curves are generated by sweeping δ.
  • RBF width γ = 0.001, 0.0025 to 0.025 in examples
    Width of Gaussian test functions in (20); affects accuracy of the dual Koopman approximation and vector field estimates.
  • Lasso regularization ρ = not reported
    Sparsity penalty in the regression (33); value must be chosen but is not specified in the experiments.
  • node function sets {φ_kl, ψ_kl} = monomials up to degree 2 in examples
    User-selected per-node basis functions; Theorem 1 requires zero mean, but examples use x^2 and u^2 which violate this.
assumptions (5)
  • domain assumption The true dynamics are exactly representable as linear combinations of known dictionary functions (24).
    All estimation is over coefficients of these dictionaries; if the dictionaries are misspecified, the identified parameters are not meaningful.
  • domain assumption Node functions satisfy zero-mean conditions (25)-(26).
    Needed for Lemma 1 and Theorem 1 orthogonality; violated in numerical experiments.
  • domain assumption The finite-dimensional EDMD and dual method yield an accurate approximation of the Koopman operator and generator on the data.
    Standard Koopman framework assumption; no error bounds are provided.
  • domain assumption The matrix logarithm (1/Ts)log(A) is well-defined for the approximated operators.
    Used in (7), (16), (43); requires A to have no nonpositive real eigenvalues, not checked in the paper.
  • domain assumption All nodes are measured and inputs are held constant over each sampling interval.
    Needed for the augmented-state construction and full network identifiability; follows from prior LTI identifiability results.

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Pith. "Pith review of Koopman operator based identification of nonlinear networks." pith.science (2026). https://pith.science/paper/LDQUXMO7

@misc{pith2026241218492,
  author       = {Pith},
  title        = {Pith review of: Koopman operator based identification of nonlinear networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LDQUXMO7}},
  note         = {Machine review of arXiv:2412.18492}
}
read the original abstract

In this work, we develop a method to identify continuous-time nonlinear networked dynamics via the Koopman operator framework. The proposed technique consists of two steps: the first step identifies the neighbors of each node, and the second step identifies the local dynamics at each node from a predefined set of dictionary functions. The technique can be used to either identify the Boolean network of interactions (first step) or to solve the complete network identification problem that amounts to estimating the local node dynamics and the nature of the node interactions (first and second steps). Under a sparsity assumption, the data required to identify the complete network dynamics is significantly less than the total number of dictionary functions describing the dynamics. This makes the proposed approach attractive for identifying large dimensional networks with sparse interconnections. The accuracy and performance of the proposed identification technique are demonstrated with several examples.

Figures

Figures reproduced from arXiv: 2412.18492 by the authors.

Figure 1
Figure 1. According to Proposition 1, the errors e and ˜e reach their minimum at the same value of the parameter γ. 4 Network Identification using Koopman Operator In this section, we aim to identify a nonlinear networked dynamics through the Koopman operator framework. Specifically, we will develop a two-step procedure where we perform the identification of the topology in the first step and the local identification of the n… view at source ↗
Figure 2
Figure 2. Effect of varying the amount of data on a network with [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Effect of varying the size of a network with data points [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Effect of varying the noise variance on a network with [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]
Figure 5
Figure 5. Figure 5: Effect of varying the edge probability ρ with N = 50, noise variance σ = 0.01, and number of data points K = 600. (a) ROC curves for several edge probability values ρ (b) ROC comparison with ρ = 0.3 and K = 250 [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: ROC curve for the Boolean reconstruction of the Erdos-Renyi network [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: Estimation of the parameters in a network with a non-polynomial vector [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Boolean reconstruction Watts-Strogatz network for different threshold [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: Predicted vs true values of parameters at node 50 [PITH_FULL_IMAGE:figures/full_fig_p028_9.png]

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