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REVIEW 2 major objections 5 minor 65 references

On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper establishes that adding a state-space projection to EDMD-DL turns a linear Koopman predictor into a nonlinear neural network model, whose continuous-time form is a neural ODE.

desk verdict A clean, useful statement that projected EDMD-DL is a neural ODE; the discrete-time equivalence is solid, the continuous-time variant is conditional on the matrix logarithm step, and the numerics need error bars. read the letter →

arxiv 2411.12940 v2 pith:HMP3SUDK submitted 2024-11-20 nlin.CD cs.LG

Jake Buzhardt , C. Ricardo Constante-Amores , Michael D. Graham This is my paper · ORCID
classification nlin.CDcs.LG
keywords Koopmanoperatorextendeddynamicmodedecompositiondictionarylearningneuralordinarydifferentialequationsstate-spaceprojectiondata-drivenpredictionchaoticdynamicsturbulentshearflow
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 is trying to show that two apparently different data-driven forecasting frameworks are actually the same once a small algorithmic change is made. Extended dynamic mode decomposition with dictionary learning (EDMD-DL) is normally a linear method on a high-dimensional space of observable functions; but when the state is projected back to the original space after every time step, the map becomes a nonlinear neural-network approximation of the flow map. In continuous time, the same projection converts the learned Koopman generator into a neural ODE. The authors argue this explains why the projection dramatically improves prediction quality on chaotic systems, and they support it by showing that the projected method matches neural ODE performance on the Lorenz system and a nine-mode turbulent shear-flow model.

What carries the argument

The load-bearing object is the projection-with-relift step, $x_{i+1}=P K_\tau^T \Psi(x_i;\theta_\Psi)$ in discrete time and $dx/dt=P L^T \Psi(x;\theta_\Psi)$ in continuous time. Here $\Psi$ is a neural-network dictionary that lifts the state to a high-dimensional feature space, $K_\tau$ is the EDMD-DL approximation of the Koopman operator, and $P$ is a linear map back to the state. This structure shows that projecting on each step reintroduces nonlinearity and makes the whole predictor a neural network with one hidden feature layer; the continuous-time version is a neural ODE whose vector field is a linear combination of learned features.

What would settle it

One could compute the learned feature map and operator from a Lorenz dataset, then integrate the continuous-time equation $dx/dt=P L^T\Psi(x)$ and compare it to the discrete rollout; a large divergence at small $\tau$ would indicate that the claimed equivalence between the two forms does not hold for the trained model.

Watch

Extended reading notes

Core claim

The central discovery is an identity between projected Koopman prediction and neural-network state-space modeling. With the dictionary $\Psi(x;\theta_\Psi)$ realized as a neural network and the state included among the dictionary elements, the one-step predictor $x_{i+1}=P K_\tau^T \Psi(x_i;\theta_\Psi)$ is exactly a nonlinear discrete-time flow-map model built by expanding the state into features and applying a linear layer. Its continuous-time counterpart, $dx/dt = P L^T \Psi(x;\theta_\Psi)$ with $L=(1/\tau)\log K_\tau$, is a neural ODE obtained directly from the EDMD-DL training, with no separate training step. The paper uses this equivalence to explain why EDMD-DL with projection succeeds where standard linear EDMD-DL fails, and to propose hybrid models that combine EDMD-DL dictionaries with neural ODE training.

Load-bearing premise

The core premise is that the learned dictionary and Koopman matrix are accurate enough that projecting back to the state at every step gives a faithful flow-map model; the continuous-time version additionally requires the matrix logarithm to recover the generator at the chosen sampling interval.

Editorial extensions

If this is right

  • EDMD-DL with projection should be classified as a nonlinear neural-network state-space model, not a linear Koopman predictor, so the linearity-based guarantees of Koopman methods do not automatically carry over.
  • On the Lorenz system and the nine-mode shear-flow model, projected EDMD-DL achieves short-time prediction error below about 3% at one Lyapunov time, comparable to neural ODEs and far better than standard linear EDMD-DL.
  • The continuous-time EDMD direct neural ODE can be constructed from the discrete-time model by a matrix logarithm without additional training, provided the sampling interval is small; at larger intervals its accuracy degrades.
  • Including the state explicitly in the dictionary separates linear and nonlinear parts of the vector field, and the resulting EDMD-structured neural ODE slightly outperforms a basic neural ODE in short-time tracking.

Reading between the lines

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

  • The equivalence suggests that any per-step projection in a Koopman-style predictor could be replaced by a trainable linear layer after the dictionary, blurring the line between physical dictionary design and generic deep learning.
  • Because the continuous-time variant hinges on the matrix logarithm, a natural extension is to train the generator $L$ directly with a velocity-matching loss rather than deriving it from $K_\tau$; this would likely extend the equivalence to larger sampling intervals.
  • The state-space form opens the door to uncertainty quantification via neural-ODE Bayesian or ensemble methods, a direction the paper does not pursue.
  • One testable consequence is that the method should work on any system whose flow map is expressible as $P K^T \Psi(x)$ for a sufficiently expressive dictionary; finding a chaotic system where no such representation exists would stress-test the claim.
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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 / 5 minor

Summary. The paper shows that EDMD-DL with a state-space projection on each rollout step defines a nonlinear discrete-time map on the state space, x_{i+1}=P K_tau^T Psi(x_i; theta_Psi), which is structurally a neural network approximation of the flow map (Eqs. 17-18). It further claims that in continuous time the same construction yields a neural ODE, dx/dt = P L^T Psi(x; theta_Psi), with L = (1/tau) log K_tau (Eqs. 21 and 25). The authors then compare six models - basic and structured neural ODEs, standard and projected EDMD-DL, an EDMD-basis neural ODE, and an EDMD direct neural ODE - on the Lorenz system and the nine-mode MFE shear-flow model, using short-time error, long-time statistics, lifetime distributions, and rare-event F-scores. The discrete-time equivalence is a clean structural result; the continuous-time claim is stated without validating the matrix-logarithm reconstruction on which it depends.

Significance. If the continuous-time caveat is properly addressed, the paper makes an important conceptual point: a Koopman-based predictor that reprojects to the state space at each step is not a linear Koopman model but a nonlinear, neural-network state-space model, and it can be interpreted as a particular neural ODE. The algebraic derivation of Eq. 18 from the definitions is sound and does not rely on any fitted-parameter assumption. The numerical comparison is unusually thorough for this literature, with Wasserstein distances for stationary densities, survival functions, and F-scores for rare events, and the architectures and training details are reported in sufficient detail to be reproduced. The principal limitations are that the continuous-time equivalence rests on the unvalidated matrix logarithm in Eq. 21, and the empirical comparisons are all single-seed runs without confidence intervals.

major comments (2)
  1. [III D, Eq. (21); Fig. 5] The continuous-time equivalence is the load-bearing step of the paper's second claim, but it is not established. Equation (21) defines L = (1/tau) log(K_tau) from the EDMD-DL matrix K_tau that was trained to minimize the one-step discrete prediction loss in Eq. (15). Nothing in that training enforces that K_tau is invertible, that its logarithm is real, or that K_tau lies near the one-parameter semigroup e^{tau L} generated by that L. The paper correctly notes that the finite-difference approximation in Eq. (20) is restricted to sufficiently small tau, but it gives no analogous condition or diagnostic for Eq. (21). Figure 5 provides the empirical signature of this gap: the EDMD direct neural ODE degrades markedly for tau >= 0.06 on the Lorenz system, while the discrete-time projected EDMD-DL remains accurate. The discrete-time equivalence in Eq. (18) is unaffected, but the continuous-time claim in Eq. (25), in the abstract, and in the conclusions should be explicitly conditioned on validation of Eq. (21), for example by checking that exp(tau L) reproduces K_tau and that P L^T Psi(x) is real.
  2. [IV B-C; abstract] The numerical support for the claims of comparable performance and for the observed degradation of the EDMD direct neural ODE in Fig. 5 comes from single training runs without confidence intervals. The Lorenz tau-sweep shows a clear qualitative trend, but with one seed it is not possible to tell whether the breakdown at tau >= 0.06 is intrinsic to the construction or specific to one optimization run; the same applies to the F-score comparison in Fig. 10 and the Wasserstein distances in Fig. 8. Because the equivalence claim is structural, this does not invalidate the paper, but the empirical comparisons should either be repeated over at least three to five seeds with error bars, or the wording equivalent performance should be softened to reflect the limited statistical evidence.
minor comments (5)
  1. [Fig. 5] The legend entries in Fig. 5 appear corrupted (for example, EDMD s ruc ured, wi h projec ion, and direc ); please regenerate the figure with clean labels.
  2. [Fig. 5] The axis label in Fig. 5 reads E(t = 2) but the text says the error is evaluated at t = 2 tau_L; the axis should be labeled accordingly.
  3. [III D, Eq. (23)] In Eq. (23), if the reconstruction map P is nonlinear, partial P / partial Psi denotes the Jacobian evaluated at Psi(x), not a constant matrix; please state this explicitly and note that for the linear reconstruction used here partial P / partial Psi reduces to P.
  4. [III C-D] The transpose convention for K_tau is a recurring source of possible confusion: Eq. (11) uses K_tau^T, while in Sec. III D the generator is defined as L = (1/tau) log K_tau and then used as L^T in dPsi/dt = L^T Psi. An explicit remark that the generator for the column-vector evolution is L^T = (1/tau) log(K_tau^T) would help readers.
  5. [IV C, Fig. 10] The comparison with Racca and Magri's F-scores appears to be read from their paper rather than recomputed; please state this explicitly so that the reader knows the two sets of scores were not obtained under identical evaluation code.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the EDMD-DL/neural-ODE equivalence is structural; the matrix-log step is a validity caveat, not a circularity.

full rationale

The derivation chain is self-contained. The discrete equivalence is an algebraic identity: with the state included in the dictionary (Eq. 14), the projected update xi+1 = P K_tau^T Psi(xi; theta_Psi) (Eq. 17) is literally a composition of a neural-network lifting, a linear matrix, and the fixed linear projection P, i.e., a neural-network flow map (Eq. 18); this is a structural classification, not a fitted quantity presented as a prediction. The continuous-time version (Eqs. 23-25) follows by the chain rule once one accepts dPsi/dt = L^T Psi, with L obtained from the matrix logarithm in Eq. 21. That step is where the paper's strongest claim becomes conditional: the authors do not verify that the learned K_tau is invertible and close to exp(tau L), and their own Fig. 5 shows the EDMD direct neural ODE degrading once tau >= 0.06. This is a correctness/assumption gap, not circularity, because the model's worth is assessed against true trajectories rather than assumed. The self-citations (Refs. 16, 41, 46, 55) support background claims or higher-dimensional applicability and are not load-bearing for the equivalence proof; no uniqueness theorem is imported. The numerical comparisons are empirical and external to the derivation. Overall no significant circularity; score 1 reflects only the mild definitional flavor of Eq. 18 and the unvalidated logarithm step, neither of which reduces a prediction to its input.

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

The structural equivalence is self-contained algebra: no free parameters are fitted to make Eq. 18 or Eq. 25 true. The numerical comparison involves many training choices (network sizes, loss weights, learning rates), but those are not part of the derivation. The main axioms are standard Koopman and EDMD assumptions plus the specific implementation choices of a linear projection and, for the direct continuous-time variant, a matrix logarithm of the learned Koopman matrix.

assumptions (4)
  • domain assumption The reconstruction map P is linear and the state x is included explicitly in the dictionary Psi, so Psi(x) = (x; tilde Psi(x)) (Eq. 14, Sec. III C).
    Linear P is required for the simplified continuous-time form in Eq. 24 and for reading Eq. 25 as a neural ODE; the authors choose this structure for all EDMD-DL implementations.
  • domain assumption The learned Koopman matrix K_tau is invertible and L = (1/tau) log K_tau (Eq. 21) is a valid generator for the continuous-time model.
    Used only for the EDMD direct neural ODE; Fig. 5 shows this construction degrades at larger sampling intervals tau >= 0.06.
  • standard math Training data is sufficiently large, noise-free, and samples the natural measure so the EDMD least-squares approximation tracks the Koopman operator.
    Stated in Sec. III before Eq. 8 and justified by convergence results of Refs. 14, 34, 35; needed for the projected model to approximate the true flow map.
  • domain assumption The neural network dictionary and linear output layer are expressive enough to represent the flow map or vector field.
    The numerical claim that projected EDMD-DL performs like a neural ODE assumes training reaches comparable representational quality; this is not proven, only observed on two systems.

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Pith. "Pith review of On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions." pith.science (2026). https://pith.science/paper/HMP3SUDK

@misc{pith2026241112940,
  author       = {Pith},
  title        = {Pith review of: On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HMP3SUDK}},
  note         = {Machine review of arXiv:2411.12940}
}
read the original abstract

This work explores the relationship between state space methods and Koopman operator-based methods for predicting the time-evolution of nonlinear dynamical systems. We demonstrate that extended dynamic mode decomposition with dictionary learning (EDMD-DL), when combined with a state space projection, is equivalent to a neural network representation of the nonlinear discrete-time flow map on the state space. We highlight how this projection step introduces nonlinearity into the evolution equations, enabling significantly improved EDMD-DL predictions. With this projection, EDMD-DL leads to a nonlinear dynamical system on the state space, which can be represented in either discrete or continuous time. This system has a natural structure for neural networks, where the state is first expanded into a high dimensional feature space followed by a linear mapping which represents the discrete-time map or the vector field as a linear combination of these features. Inspired by these observations, we implement several variations of neural ordinary differential equations (ODEs) and EDMD-DL, developed by combining different aspects of their respective model structures and training procedures. We evaluate these methods using numerical experiments on chaotic dynamics in the Lorenz system and a nine-mode model of turbulent shear flow, showing comparable performance across methods in terms of short-time trajectory prediction, reconstruction of long-time statistics, and prediction of rare events. These results highlight the equivalence of the EDMD-DL implementation with a state space projection to a neural ODE representation of the dynamics. We also show that these methods provide comparable performance to a non-Markovian approach in terms of prediction of extreme events.

Figures

Figures reproduced from arXiv: 2411.12940 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic illustrating how EDMD-DL with a projection to the state space directly leads to a neural network repre [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic illustrating how the continuous-time EDMD-DL formulation with projection to the state space leads to a [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Prediction results for the data-driven models on the Lorenz system for sampling interval [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (7 more)
Figure 5
Figure 5. Figure 5: We see that at large sampling times, the EDMD [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Long time predictions of each model from a common initial condition for 100 time units on the Lorenz system. Each [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Normalized ensemble-averaged error of each model [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Prediction results for the data-driven models on the MFE model of a turbulent shear flow. (a) Normalized ensemble [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Typical time series for the energy in the MFE model [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Joint probability densities of amplitudes [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Lifetime distribution predictions for the models, given [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

Discussion (0). Continue with ORCID to comment.

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