{"id":"ebbe0e67-e7b3-4f02-baec-4cf695bcaba5","arxiv_id":"2411.12940","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Projected EDMD-DL is structurally a neural ODE: it lifts the state through a learned dictionary, evolves or differentiates with a linear map, and projects back; on Lorenz and a nine-mode shear flow it performs comparably to directly trained neural ODEs.","lead":"The paper shows that EDMD-DL, a Koopman-based prediction method, becomes a neural network model of the flow map when it projects back to the state space at every step, which is the same structure as a neural ODE. This explains why projected Koopman predictions beat linear Koopman evolution on chaotic systems and gives practitioners a unified view of two popular methods.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Continuous-time equivalence (Eq. 25) hinges on matrix logarithm (Eq. 21); the paper never checks whether learned K_tau admits a real generator, and Fig. 5 shows this variant fails for tau >= 0.06. Discrete Eq. 18 stands, but the continuous-time claim is conditional on an unvalidated step.","rationale":"The reader's weakest assumption identifies exactly the same bottleneck: the continuous-time direct neural ODE depends on the matrix logarithm in Eq. 21 being a faithful generator, which is not guaranteed by the EDMD-DL training objective and is contradicted by the paper's own Fig. 5 at larger sampling intervals. My reading confirms this. I do not see a deeper flaw in the discrete-time equivalence of Eq. 18: with the state explicitly in the dictionary, P K_tau^T Psi(x; theta_Psi) is literally a neural network representation of the one-step flow map, and the experiments are consistent with that interpretation. The concern is therefore not fatal to the paper's main structural observation, but it does mean the continuous-time half of the claim ('represented in either discrete or continuous time') is not robust without a validation of the logarithm step. This supports the reader's CONDITIONAL verdict rather than moving it; I would keep the requirement that the authors either restrict Eq. 25 to regimes where the matrix logarithm is verified or add a diagnostic for its validity. I also note the paper flags the small-tau limitation for the finite-difference generator (Eq. 20) but not for the matrix-log generator (Eq. 21), which is an omission worth correcting. No code or data are released, so the empirical comparison cannot be independently checked; this is secondary to the mathematical concern but reinforces the conditional status.","tokens_in":21632,"tokens_out":12403,"duration_ms":131788,"concrete_test":"For the same trained EDMD-DL checkpoints used in Fig. 5 (tau = 0.02, 0.06, and 0.1), compute L = (1/tau) log K_tau with a real-matrix logarithm routine and record whether a real L exists. Then, on 1000 test states, compare the discrete projected map P K_tau^T Psi(x) with the numerical flow over [0, tau] of the ODE dx/dt = P L^T Psi(x), using the same ODE solver as in the paper, and report the normalized mismatch delta(tau) = mean_x ||flow(x) - P K_tau^T Psi(x)|| / mean_x ||x||. If delta is small at tau = 0.02 but jumps at tau >= 0.06 (or if the real logarithm fails), Eq. 21 is the limiting assumption and the continuous-time equivalence claim should be narrowed; if delta remains small while trajectory errors degrade, the bottleneck is instead long-time integration drift.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim has two parts. The discrete part, Eq. 18, is definitionally sound: with the state included in the dictionary, P K_tau^T Psi(x; theta_Psi) is a neural network flow map, and the numerical comparison supports that interpretation. The load-bearing weakness is the continuous-time part, Eq. 25. To obtain L, the paper uses L = (1/tau) log K_tau (Eq. 21). For this to be a faithful generator matrix, K_tau must be invertible, its logarithm must be real (or at least yield a real vector field after multiplication by P), and K_tau must lie near the one-parameter semigroup generated by that L. The EDMD-DL objective (Eq. 15) optimizes a one-step discrete prediction loss and imposes none of these conditions. The paper explicitly warns that Eq. 20 is accurate only for sufficiently small tau, but gives no analogous condition or diagnostic for Eq. 21. Figure 5 is the empirical signature of this gap: the EDMD direct neural ODE constructed via Eq. 21 degrades markedly for tau >= 0.06, while methods not relying on the logarithm remain accurate. Thus the claim that projected EDMD-DL 'can be represented in continuous time directly' is only as secure as the logarithmic reconstruction step, and the paper does not establish when that step is valid. The discrete equivalence and the practical comparison survive, but the continuous-time claim should be explicitly conditioned on a validation of Eq. 21.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":1457,"tokens_out":1514,"duration_ms":110120,"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":[{"comment":"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.","section":"III D, Eq. (21); Fig. 5"},{"comment":"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.","section":"IV B-C; abstract"}],"minor_comments":[{"comment":"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.","section":"Fig. 5"},{"comment":"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.","section":"Fig. 5"},{"comment":"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.","section":"III D, Eq. (23)"},{"comment":"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.","section":"III C-D"},{"comment":"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.","section":"IV C, Fig. 10"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the central discrete-time observation is valuable. The main revision should focus on the continuous-time construction: either add a concrete validation of Eq. (21) or restrict the claim to discrete time. The authors cite several of their own prior papers (Refs. 16, 41, 46), but the central derivation is independent of those results, so I do not see a circularity problem. Releasing code for the numerical experiments would also strengthen reproducibility for a methods-oriented paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper's main claim is the equivalence between EDMD-DL with state projection and neural ODEs. The discrete-time identity (Eq. 18) is almost a one-line observation once the dictionary is a neural net, but it is stated clearly and it is genuinely useful: projected EDMD-DL is not a linear Koopman predictor, it is a nonlinear flow-map approximator. The continuous-time form (Eq. 25) is also derived cleanly, and the paper is honest that it relies on a generator matrix L obtained from the matrix logarithm of K_tau. That step is the soft spot. The EDMD-DL objective does not constrain K_tau to be invertible or its logarithm to be real, and Fig. 5 shows the direct neural ODE degrades badly for tau >= 0.06. The discrete equivalence survives; the continuous-time claim should be conditioned on a validity check of the logarithm, which the paper doesn't provide.\n\nThe numerical comparisons are reasonable and support the main point: projected EDMD-DL performs like the neural ODE variants on the Lorenz system and the MFE shear-flow model, while linear EDMD-DL fails. But the experiments are single-seed, lack confidence intervals, and no code or data are released, so the empirical equivalence is not yet robust. The comparison to Racca and Magri's non-Markovian model is a nice addition, though it is only one test case.\n\nOn the citation pattern: several supporting references are from the same group (16, 41, 46), but the central derivation is algebraic and does not depend on those results. The paper gives prior work credit where it is due, especially van Goor et al.'s manifold projection view.\n\nWho is this for? Anyone working on Koopman-based forecasting or neural ODEs for dynamical systems. It gives a common language and explains why projection helps so much. It is not a new predictive algorithm; it is a conceptual clarification with supporting experiments. That is a legitimate contribution.\n\nRecommendation: send it out. It deserves serious review, mainly to push the authors to add uncertainty estimates, release code and data, and either fix or explicitly condition the continuous-time claim. I would review it myself, or hand it to someone who knows both literatures.","headline":"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.","tokens_in":22495,"tokens_out":1916,"would_cite":true,"duration_ms":19017,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Koopman operator","extended dynamic mode decomposition","dictionary learning","neural ordinary differential equations","state-space projection","data-driven prediction","chaotic dynamics","turbulent shear flow"],"falsifier":"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.","tokens_in":21398,"feed_emoji":"🔄","tokens_out":6139,"duration_ms":58327,"temperature":0.7,"pith_summary":"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.","feed_headline":"Projected Koopman predictors are neural ODEs in disguise","feed_subtitle":"Adding a per-step state projection makes EDMD-DL a nonlinear neural network, matching neural ODE accuracy on chaotic flows.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the EDMD-DL framework: dictionary as neural network, joint optimization of dictionary and Koopman matrix.","marker":"[2]"},{"why":"Defines neural ODEs and their training; the counterpart that projected EDMD-DL is shown to match.","marker":"[1]"},{"why":"Provides EDMD, the least-squares Koopman approximation underlying the dictionary learning objective.","marker":"[14]"},{"why":"Shows projection during rollout improves EDMD-DL predictions; the direct predecessor of this paper's equivalence.","marker":"[16]"},{"why":"Demonstrates effectiveness of state-space projection for EDMD forecasting, part of the motivation.","marker":"[21]"},{"why":"Gives manifold/projection interpretation of relifting, supporting the structural reading of the projection step.","marker":"[22]"},{"why":"Provides the matrix-logarithm route from Koopman operator to generator used in Eq. 21.","marker":"[33]"},{"why":"Defines the nine-mode MFE shear-flow model used as the extreme-event test case.","marker":"[56]"}],"fun_headline_variants":["Projected Koopman is a neural ODE in disguise","Projection turns Koopman into a neural network","Projected EDMD-DL equals a neural ODE","Koopman with projection matches neural ODEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Projected Koopman is a neural ODE in disguise","Projection turns Koopman into a neural network","Projected EDMD-DL equals a neural ODE","Koopman with projection matches neural ODEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000679,"raw_usage":{"total_tokens":3123,"prompt_tokens":1019,"completion_tokens":2104,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":635,"completion_tokens_details":{"reasoning_tokens":2038}},"tokens_in":635,"tokens_out":2104,"duration_ms":15295,"temperature":1.0,"reasoning_tokens":2038,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:01:31.336366+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines neural ODEs and their training; the counterpart that projected EDMD-DL is shown to match."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows projection during rollout improves EDMD-DL predictions; the direct predecessor of this paper's equivalence."},{"cited_title":"Junker , author J","cited_arxiv_id":null,"evidence_quote":"Demonstrates effectiveness of state-space projection for EDMD forecasting, part of the motivation."}],"review_version":1}