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REVIEW 4 major objections 5 minor 27 references

Data-driven Koopman Operator-based Prediction and Control Using Model Averaging

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Weighted Koopman ensemble beats single-model prediction and control.

desk verdict A clean but thinly evidenced model-averaging recipe for Koopman embeddings; worth reviewing, but the generalization claim is contradicted by the paper's own validation numbers. read the letter →

arxiv 2412.02984 v1 pith:IOA7DZZN submitted 2024-12-04 math.OC

Daisuke Uchida , Karthik Duraisamy This is my paper · ORCID
classification math.OC MSC 93C1093C5537M9962F15
keywords KoopmanoperatormodelaveragingBayesianneuralnetworkfeaturesextendeddynamicmodedecompositionpredictivecontrolnonlinearsystemslinearembedding
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 proposes Koopman Model Averaging (KMA), a data-driven way to build a Koopman-operator linear embedding model for nonlinear control systems that is more accurate and generalizes better than any single model trained on the same data. It trains one neural-network feature map on a first data subset, fits a small ensemble of Extended Dynamic Mode Decomposition models—each a linear embedding of the form $z_{k+1}=A_i z_k+B_i u_k$—on separate subsets, and then averages their parameters using Bayesian Model Averaging weights computed from holdout data. The result is a single weighted linear embedding model, equations (28)–(29), whose predicted next state is a posterior-weighted sum of the ensemble's outputs. In numerical tests on the Duffing oscillator and the cartpole, KMA tracks references and predicts states at least as well as, and in several cases better than, both a standard EDMD model and a normal neural-network model, even when the single model has lower validation loss. The authors' claim is that this weighting scheme compensates for each ensemble member's errors across different operating regimes, giving a practical surrogate for both prediction and controller design.

What carries the argument

The central object is the weighted linear embedding model of Equations (28)–(29), whose parameters are Bayesian-Model-Averaging-weighted sums of the ensemble members' parameters. The key mechanism is that the posterior expectation of the next embedded state, $\mathbb{E}[z_{k+1}\mid D]=\sum_i w_i(A_i z_k+B_i u_k)$, preserves the form of a single linear time-invariant system, so the averaged model still supports linear LQR and MPC designs. The feature map $g(\cdot;\theta_g)$ learned once by a neural-network base model on subset $D_1$ is shared by all ensemble members; the remaining members are obtained by closed-form EDMD fits on data subsets, and the weights $w_i$ are approximated by an AIC-type pseudo-BMA (expected log pointwise predictive density) computed on a holdout set.

What would settle it

Run the same Duffing and cartpole experiments but train the base model on a narrow slice of the state space (for example, initial conditions only in $[0,1]^2$, with the ensemble subsets drawn from the full $[-3,3]^2$ range) so that the shared $g$ is demonstrably poor outside the base region. If the weighted model still beats the single normal neural-network model on the full-range test data, the shared-feature assumption is not the limiting factor; if KMA degrades to the level of the individual EDMD fits, that would show the result hinges on having a good base feature map.

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Extended reading notes

Core claim

The central claim is that a weighted linear embedding model, formed as $z_{k+1}\approx(\sum_i w_i A_i)z_k+(\sum_i w_i B_i)u_k$ and $x^{\mathrm{pred}}_{k+1}=(\sum_i w_i C_i A_i)z_k+(\sum_i w_i C_i B_i)u_k$, where the weights $w_i$ approximate posterior model evidence, is a better and more generalizable predictor of nonlinear controlled dynamics than any single Koopman-based model in the ensemble. The paper shows that applying Bayesian Model Averaging to point estimates of the embedded state and the original state collapses the ensemble back into the same linear embedding structure used for LQR and MPC, so the model-uncertainty-aware average costs no additional control-design complexity. The Duffing oscillator and cartpole experiments support the claim: KMA matches or outperforms both an EDMD model and a neural-network-trained single model in state prediction, LQR stabilization, and reference-tracking MPC, despite the base model used in KMA having a larger validation loss than the normal neural-network model, which the authors interpret as better behavior on unseen operating regimes.

Load-bearing premise

The whole method depends on the single feature map $g$ learned by the base model on the first data subset being a good enough embedding for all the operating regimes the other ensemble models are fit on; if that one map is poor outside the base training region, the later weighted-average fits cannot repair it, and the paper does not test how sensitive KMA is to this feature-map choice.

Editorial extensions

If this is right

  • Because the averaged model is itself linear time-invariant, any LQR, linear MPC, or other linear controller design can be applied directly to it; controller synthesis needs no change relative to single-model Koopman approaches.
  • The method turns a single expensive neural-network training into one base training plus several cheap closed-form EDMD fits, so improving generalization through ensembling does not require repeatedly solving the non-convex training problem.
  • The weighted parameters remain interpretable: each ensemble member contributes according to its posterior evidence on holdout data, which gives a principled way to discard or down-weight poorly performing models.
  • On the two tested systems, a model with a worse validation loss (the KMA base model) outperformed a better-fit single model in reference tracking and long-horizon prediction, indicating that validation loss on a finite data set is not a reliable proxy for closed-loop control performance.

Reading between the lines

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

  • The success of KMA likely depends on the holdout set $D_a$ being representative of where the model will be used, since the pseudo-BMA weights are estimated on that set; a deployment distribution that shifts away from $D_a$ would require recomputing weights.
  • The same parameter-averaging recipe could be applied to bilinear Koopman embeddings or to models with nonlinear decoders, since only the linear-in-the-parameters structure is used in the derivation; this is a direct extension the paper does not test.
  • A natural next experiment is to compare KMA against other ensemble combination rules (plain averaging, stacking, or weights based on validation error) on the same benchmarks to isolate how much of the gain comes from Bayesian weighting versus from having multiple EDMD fits with a shared feature map.
  • The paper reports only point estimates of the posterior; extending to full predictive covariance would enable uncertainty-aware MPC, which would use the same Gaussian assumptions that are already stated.
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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

4 major / 5 minor

Summary. The paper proposes a model-averaging method for Koopman operator-based modeling of unknown nonlinear control systems. A base neural-network Koopman model is trained on a subset of the data to obtain feature maps; an ensemble of linear embedding models is then fitted on different subsets via EDMD, and Bayesian Model Averaging (BMA) weights, computed on a calibration set D_a, are used to form a weighted linear model (Eqs. (28)-(29)). The authors claim that this weighted model has better state-prediction accuracy and generalizability than a single model, and they support the claim with two numerical examples (Duffing oscillator and cartpole) covering prediction, LQR, and MPC.

Significance. If the empirical claims held, the paper would present a simple, modular way to combine multiple Koopman models into a single linear embedding model while remaining model-uncertainty aware. The derivation of the weighted model from BMA is elementary and correct, the algorithm is explicitly stated, and the idea of averaging in the embedding space is reasonable. However, the evidence for the central generalization claim is thin: only two examples, single hand-selected trajectories, no error bars or statistical analysis, and in both examples the reported validation loss of the proposed model is worse than that of the normal NN baseline. The significance is therefore conditional on a substantially strengthened empirical evaluation.

major comments (4)
  1. [Section V-B, Fig. 2b] The central claim that the proposed weighted model has better predictive capability and generalizability than a single model is not supported by the paper's own quantitative results. In the cartpole example, the validation loss of the proposed model is 2.82e-5, which is higher than the normal NN model's 1.16e-5, and the claimed advantage rests entirely on one hand-selected trajectory (IC 2) with no error bars, multiple seeds, or rollout statistics. The authors should report prediction errors (e.g., mean and standard deviation over many initial conditions) and establish statistical significance rather than relying on a single favorable trajectory.
  2. [Section V-A and Algorithm 1, Step 5] The comparison between the proposed method and the baselines is confounded by the role of the data set D_a. The normal NN and EDMD baselines are trained on the entire data set D = (∪_i D_i) ∪ D_a, while the proposed KMA uses D_a only to compute the BMA weights (Algorithm 1, Step 5). Thus D_a acts as a validation/calibration set for KMA but as a training set for the baselines, making the reported validation losses and closed-loop comparisons not apples-to-apples. The authors should either train the baselines only on ∪_i D_i and use a separate held-out test set for all methods, or clearly define and use a test set that no method has seen during any stage of training or weight computation.
  3. [Section IV-A, Eq. (19)] The computation of the BMA weights is under-specified, which is load-bearing because the weighted model (28)-(29) is determined entirely by these weights. The text states only that 'PyMC is used to compute w_i' but does not specify the likelihood model, the priors, or how the Gaussian assumptions in Eqs. (24)-(25) are used, nor how the integrals in Eq. (16) are approximated. Without these details, the method is not reproducible. Please provide the exact probabilistic model, the estimation procedure (e.g., the LOO approximation used), and all relevant hyperparameters.
  4. [Section V-A, Fig. 1e and Section V-B, Fig. 2b] The claimed superiority of the proposed model over the normal NN model is based on qualitative observations of single closed-loop or prediction trajectories. In the Duffing example, no validation loss for the actual weighted model is reported—only that of the base model—so the generalization claim for KMA is unquantified; the single MPC trajectory (Fig. 1e) is not enough to establish the advantage. The authors should provide quantitative performance metrics (e.g., tracking error, state prediction RMSE over an ensemble of initial conditions, and closed-loop cost) and, for Duffing, report the weighted model's validation loss as well.
minor comments (5)
  1. [Section II, Eq. (12)] There is a typo in Eq. (12): the virtual system should be written as ξ_{k+1} = A ξ_k + B u_k, not ξ_{k+1} = A ξ_{k+1} + B u_k.
  2. [Section IV-A, Eq. (19)] The expression for elpd_i is confusing: the integral over the true distribution p_t(\tilde{q}_j) is not the quantity that is evaluated in practice, and the notation \tilde{q}_j is not defined clearly. Please rewrite Eq. (19) using the usual definition of the expected log pointwise predictive density or the LOO approximation used to compute it.
  3. [Section IV-A] The phrase 'posterior model evidence' for p(M_i | D) is imprecise; Eq. (15) gives the posterior model probability. The marginal likelihood in Eq. (16) is the model evidence.
  4. [Section V-A] The discretization used to obtain the difference equation from the continuous Duffing dynamics is not stated; the text says only that a first-order discretization is used with Δt = 0.01. Please give the explicit update formula.
  5. [Section V-A and V-B] The numerical setup is missing several training details needed for reproducibility: the values of the loss weights λ1 and λ2, the optimizer, learning rate, number of epochs, batch size, and random seeds. Please include these details or a pointer to released code.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: KMA is a standard BMA-weighted ensemble whose construction is explicit and whose predictions are not used to fit the model.

full rationale

The derivation chain in Section IV is self-contained. The weighted model (28)-(29) is obtained by taking the expectation in (18) under the Gaussian assumptions (24)-(25), with weights w_i computed by pseudo-BMA (19) on a calibration subset D_a. This is a definitional statistical construction, not a circular derivation: no equation uses the evaluation targets or the reported test trajectories to define the model. The shared feature map g is learned on D_1 and then fixed, and the ensemble matrices are EDMD fits on disjoint subsets D_i; the weighted parameters are convex combinations of those fits. The claim that KMA generalizes better is an empirical assertion, and the paper's own reported validation losses (1.16e-5 vs 2.82e-5 in cartpole; 7.60e-6 vs 1.81e-5 for the base model in Duffing) actually undercut it. However, weakness of evidence or mismatch between claim and numbers is a correctness/evidence concern, not circularity. Self-citations ([6], [10], [11], [19]) are contextual references and are not load-bearing in the derivation; no uniqueness theorem or ansatz is imported from the authors' prior work to force the result. Therefore the circularity score is 0.

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

The construction depends on three things the paper takes from outside: a feature map expressive enough for all models, data-based weights that remain valid on test conditions, and the standard Koopman assumption of a near-invariant finite-dimensional subspace. The Gaussian assumptions in (24)-(25) are not needed for the weighted averages and are essentially decorative. No new physical entities are introduced.

free parameters (4)
  • Loss weights lambda1, lambda2 = not reported
    Hyperparameters in the base model loss (13); values are not reported, yet they shape the shared feature map g and every downstream model.
  • BMA weights w_i = not reported
    Pseudo-BMA and LOO weights computed on Da in Algorithm 1, Section IV; values are not reported. They define the final weighted model (28)-(29).
  • Ensemble size N and subset sizes = N=5; 100 trajectories per Di; 50 trajectories in Da
    Chosen by hand without sensitivity analysis; the method's advantage may depend on these choices.
  • Neural network architecture = 10 neurons, one hidden layer (Duffing); two hidden layers (cartpole)
    Architecture controls expressiveness of the shared feature map; no justification or ablation is provided.
assumptions (4)
  • domain assumption The dynamics (1) is unknown, but the data set D is drawn from it and is representative of the operating conditions where the model will be used.
    Assumed throughout Sections II and V; if Da and the test conditions are not representative, the BMA weights and the final model will not generalize.
  • domain assumption The chosen feature map g spans an approximately invariant subspace for the control dynamics, so the linear embedding model (9) is a valid surrogate.
    Equations (4)-(7) require invariance or approximate invariance; the paper uses this standard Koopman assumption without quantifying the approximation error.
  • ad hoc to paper The conditional distributions of z_{k+1} and x_{k+1} given each model are Gaussian with common covariances Sigma_z and Sigma (Eqs. (24)-(25)).
    Used to justify point estimates via expectation (18); it is stronger than needed, since (26)-(27) follow from linearity of expectation alone.
  • domain assumption AIC-type pseudo-BMA weights with LOO approximate the posterior model evidence p(M_i | D) in Eq. (15).
    Adopted in Section IV-A following [25]; if the approximation is poor, calling the result Bayesian model averaging is mainly a label.

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Cite this review

Pith. "Pith review of Data-driven Koopman Operator-based Prediction and Control Using Model Averaging." pith.science (2026). https://pith.science/paper/IOA7DZZN

@misc{pith2026241202984,
  author       = {Pith},
  title        = {Pith review of: Data-driven Koopman Operator-based Prediction and Control Using Model Averaging},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IOA7DZZN}},
  note         = {Machine review of arXiv:2412.02984}
}
read the original abstract

This work presents a data-driven Koopman operator-based modeling method using a model averaging technique. While the Koopman operator has been used for data-driven modeling and control of nonlinear dynamics, it is challenging to accurately reconstruct unknown dynamics from data and perform different decision-making tasks, mainly due to its infinite dimensionality and difficulty of finding invariant subspaces. We utilize ideas from a Bayesian inference-based model averaging technique to devise a data-driven method that first populates multiple Koopman models starting with a feature extraction using neural networks and then computes point estimates of the posterior of predicted variables. Although each model in the ensemble is not likely to be accurate enough for a wide range of operating points or unseen data, the proposed weighted linear embedding model combines the outputs of model ensemble aiming at compensating the modeling error of each model so that the overall performance will be improved.

Figures

Figures reproduced from arXiv: 2412.02984 by the authors.

Figure 1
Figure 1. Duffing oscillator. as a result of too simple feature map design for the four dimensional dynamics of the cartpole. The results of the state prediction are shown in Figs. 2a and 2b. The EDMD model also has difficulty in this task. Both the normal NN and the proposed models show reasonable predictions with the initial condition IC 1 (Fig. 2a). However, the prediction of the normal NN model starts deviating from the t… view at source ↗
Figure 2
Figure 2. Cartpole system. predictions that are accurate enough for a wide range of operating points even if the model possesses high accuracy w.r.t. a certain regime of dynamics such as training data, the proposed method first trains a base model with the use of neural networks and then populates an ensemble of models on different data points. Based on the ideas of the Bayesian model averaging, these models are merged into a… view at source ↗

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