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REVIEW 5 major objections 7 minor 64 references

Deep Koopman operator framework for causal discovery in nonlinear dynamical systems

T0 review · 5 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Kausal combines deep Koopman operators with causal discovery: learned observables make the causal signal between cause and effect measurable in nonlinear dynamical systems.

desk verdict A useful deep-Koopman causal discovery tool with a plausible empirical story, but the causal measure is a heuristic and the significance test doesn't control for model capacity. read the letter →

arxiv 2505.14828 v1 pith:4AGBZPQG submitted 2025-05-20 cs.LG

Juan Nathaniel , Carla Roesch , Jatan Buch , Derek DeSantis , Adam Rupe , Kara Lamb , Pierre Gentine This is my paper · ORCID
classification cs.LG
keywords causaldiscoveryKoopmanoperatordeeplearningdynamicalsystemsENSOdynamicmodedecompositionreproducingkernelHilbertspacenonlineardynamics
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 proposes Kausal, an algorithm that performs causal discovery in nonlinear dynamical systems by lifting the system into a space of observables where the dynamics are approximately linear. The central claim is that learning these observable functions with deep neural networks, rather than fixing them in advance, yields a more accurate and meaningful causal measure. The causal measure compares two Koopman models of the effect variable: one that evolves the effect's own observables, and one that also includes the proposed cause's observables; if the joint model predicts the effect's future better, the cause is said to causally influence the effect. The paper demonstrates this in coupled Rössler oscillators, a reaction-diffusion system, a conceptual ENSO model, and ocean reanalysis data, where the causal measure tracks major El Niño and La Niña events.

What carries the argument

The load-bearing object is the finite-rank Koopman approximation built from learned observables. A neural encoder produces dictionary functions $\psi_\theta$; dynamic mode decomposition fits the linear operator $K^t := \Psi^t_\theta \Psi^\dagger_\theta$; and the causal score is the squared-error gap between the marginal model $K^t_{\mathrm{marg}}$ and the joint model $K^t_{\mathrm{joint}}$ when predicting the effect's next state. The learned observables matter because the finite-dimensional dictionary must capture enough of the nonlinear dynamics for the error gap to be interpretable as causation.

What would settle it

Take a system with no coupling (e.g., the coupled Rössler equations with $c_1=c_2=0$) and run Kausal with a joint model whose encoder has many more parameters than the marginal encoder; if $\Delta K^t_{C,E}$ remains above the shuffle-based significance threshold for a non-causal direction, the causal measure is picking up model capacity rather than causation.

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Extended reading notes

Core claim

On its own terms, Kausal establishes that causal influence from a component $C$ to a component $E$ of a dynamical system can be read off as the difference in one-step prediction error between a marginal Koopman model built from effect observables and a joint model built from effect-plus-cause observables, $\Delta K^t_{C,E} = L(\tilde{\omega}^t_{E|\mathrm{marg}}, \omega^t_E) - L(\tilde{\omega}^t_{E|\mathrm{joint}}, \omega^t_E)$. The authorial claim is that the quality of this causal signal hinges on the observables: neural-network-parameterized dictionaries $\psi_\theta$ infer the basis functions that make the finite-rank approximation of the Koopman operator accurate, whereas prescribed dictionaries such as random Fourier features misestimate or miss the signal. In the experiments the true direction yields a positive $\Delta K^t$ that is statistically distinct from a time-shuffle null, while the non-causal direction does not, and in the observed ENSO record the inferred causal measure rises during major El Niño events.

Load-bearing premise

The method assumes that any reduction in held-out prediction error when the cause's observables are added to the effect's model is caused by the cause, rather than by the joint model's larger input or greater flexibility.

Editorial extensions

If this is right

  • In the tested systems, learning observables with MLPs or CNNs recovers statistically significant causal signal in the true direction while the reverse direction stays at the shuffle null, and this separation is weaker or absent with prescribed RFF observables.
  • Causal magnitude becomes a time-varying quantity: the ENSO causal measure grows at major El Niño and La Niña events, suggesting the method can monitor coupling strength in real-world records.
  • Ensembling randomly initialized neural dictionaries yields a spread of causal measures, providing an uncertainty estimate for the inferred signal.
  • On sliding-window AUROC benchmarks against PCMCI+, VARLiNGAM, cLSTM, and TSCI, Kausal achieves the highest detection scores for causal extremes in the Rössler, reaction-diffusion, and ENSO tasks.

Reading between the lines

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

  • Because the causal score is a predictive-advantage gap, a fair reader should expect that a joint model with more parameters or richer inputs could inflate $\Delta K^t$ even without causation; a capacity-matched control or a permutation test on the cause's information content would tighten the interpretation beyond the time-shuffle null.
  • The same machinery can be pointed at causal graph discovery: pairwise Kausal tests on all component pairs, combined with the multipartite extension noted in the paper, could yield adjacency estimates for nonlinear networks with time-varying edge strengths.
  • Testable extension: on a system with injected confounding or measurement noise, compare Kausal's inferred direction against a known ground-truth graph to see whether learned observables separate causal coupling from mere statistical dependence.
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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

5 major / 7 minor

Summary. The paper proposes Kausal, a causal discovery algorithm for nonlinear dynamical systems that learns Koopman observables with neural networks (MLP or CNN) and defines a causal measure as the difference in prediction error between a marginal DMD model (effect only) and a joint DMD model (effect plus candidate cause). The method is demonstrated on coupled Rössler oscillators, a nonlinear reaction-diffusion equation, a conceptual ENSO model, and ECMWF ocean reanalysis. The authors benchmark against PCMCI+, VARLiNGAM, cLSTM, and TSCI and report higher AUROC for causal magnitude detection. The code is released.

Significance. Kausal addresses an important problem: causal discovery in high-dimensional nonlinear dynamical systems, where standard Granger-type methods are limited. The combination of deep-learned observables with Koopman operator theory is timely and could be practically useful for climate and other spatiotemporal data. The paper's strengths include controlled simulations with known coupling, an ensemble-uncertainty analysis, a null-hypothesis test, and open-source code. However, the causal measure is a heuristic without a formal identification guarantee, and the empirical comparisons contain potential confounds. The central idea is promising but the evidence is not yet conclusive, so the paper requires substantive revision before the claims can be accepted.

major comments (5)
  1. [Section 3.2, Eq. (17) and Algorithm 1] The causal measure in Eq. (17) is not identified as a causal influence because the joint model differs from the marginal model not only by including the cause observables but also by having higher input dimensionality and a separately learned, higher-capacity encoder. Since K̂ is a least-squares pseudo-inverse (Eq. 8), adding features reduces in-sample error and can reduce held-out error through chance correlations or smoothness of the added inputs, even without any coupling. The null distribution in Algorithm 1 does not control for this: it randomly permutes both Ω_C and Ω_E, destroying the temporal structure of both series, so the null is not the 'same system without coupling.' A non-causal control that preserves the autocorrelation and dimensionality of the cause (e.g., phase-randomized surrogates or an independent but auto-correlated feature set) is required to support the claim that the observed asymmetry in Figures 3–6 and the ENSO tracking in Figure 9 reflect causal influence rather than predictability gains.
  2. [Section 2.2 and Appendix C] The training objective for the deep observables ψ_θ is never specified. The paper states that the dictionary functions are learned with MLP/CNN encoders and lists architectures, optimizers, and epochs in Appendix C, but it does not define the loss that is minimized during training (e.g., DMD reconstruction error, autoencoder loss, or a combination). This omission makes the core 'deep' component of Kausal irreproducible and prevents an assessment of whether the learned observables are actually optimized for the causal measure. The authors should provide the exact loss function and state whether the marginal and joint encoders are trained independently or jointly.
  3. [Section 4.5 and Appendix D.2] The baseline comparisons are not on equal footing. In the reaction-diffusion experiment, the baselines are run on a spatially averaged time series because 'most baseline algorithms do not scale in high-dimensional, multivariate nodes,' while Kausal uses the full 16×16 grid with each grid cell as an independent feature, giving Kausal access to a much richer input representation. This discrepancy can inflate Kausal's apparent advantage in the AUROC comparisons. Furthermore, in Table 1b, PCMCI+ detects the false direction strongly (ρ_E→C = 1.95×10^-13) and the Kausal RFF variant yields a significant p-value for the false direction (ρ = 0.03), which contradicts the text's claim that 'Kausal captures the true (and the lack of) causal relationship with greater precision.' These issues need to be addressed before the comparative claims can be accepted.
  4. [Section 4.4 and Figure 9] The real-world ENSO experiment does not validate causal direction. The reported positive causal measure tracks major El Niño events, but this is essentially a predictability result: including SST as an input improves prediction of sea surface height in a data-driven model even if there is no causal mechanism in the claimed direction. The paper calls these 'preliminary results,' yet the abstract and introduction present them as evidence of 'applicability to real-world phenomena.' A clear statement of the expected causal hypothesis and a non-causal control (e.g., using a different variable with similar autocorrelation as the candidate cause) is needed before the ENSO event tracking can be interpreted as causal evidence.
  5. [Section 3.2, Eqs. (17)–(19)] The definition of the causal measure is ambiguous when combined with the 'conditional forecasting' procedure of Eq. (19). In Eq. (19), the predicted output ω̃_E is fed back as the identity observable while the non-identity observables are still computed from the test data; it is not clear whether the loss L in Eq. (18) is evaluated against the realized ω_E^t at each step or against the iterated prediction, or over which data points. This matters because Figures 3, 4, and 9 all report causal measures that depend on this procedure. Please clarify the exact error aggregation used to produce the reported values.
minor comments (7)
  1. [Algorithm 1] The procedure 'PTest' is not defined; please specify whether a one-sided permutation test or a t-test is used, and state the exact statistic.
  2. [Table 1b] The Kausal RFF result (ρ = 0.03 in both directions) is not discussed; this weakens the claim that learnable observables are strictly better than prescribed ones in all settings and should be acknowledged.
  3. [Section 4.4] The use of the Nino-3 region for T and Nino-4 for h is nonstandard; please justify this choice or use the conventional Nino3.4 region for both variables to improve comparability with the ENSO literature.
  4. [Eq. (18) and Algorithm 1] The symbol N is used both for the number of data points in the loss and for the number of permutations in Algorithm 1; please use distinct notation to avoid ambiguity.
  5. [Appendix C.2] The time step for the reaction-diffusion simulation is given as Δt = 10^3, which is likely a typo for 10^-3; please correct.
  6. [Section 4.2, Eq. (21)] The sentence 'The states (u, v) include components of the horizontal velocity' is inaccurate for a reaction-diffusion system; u and v are concentration fields, not velocity components.
  7. [Section 3.2] The paper relies on Theorem III.2 of Rupe et al. (2024) for the equivalence of dynamical and Koopman causality without restating its assumptions; since this theorem is the theoretical backbone of the method, the main text should summarize the conditions under which the equivalence holds.

Circularity Check

1 steps flagged · score 4.0 of 10

Kausal's causal interpretation of its DMD error gap rests on a same-author theorem (Rupe et al., 2024) rather than a derivation here; the empirical direction tests are otherwise independent.

  1. self citation load bearing [Section 3.2, 'Koopman causal influence' (around Eq. 13); background in Section 2.1 follows Rupe et al. (2024)]
    "In summary, dynamical causality and Koopman causality at time t turn out to be equivalent. Namely, ΩC →t ΩE if and only if ΩC →t_K ΩE. The proof of this can be found in Theorem III.2 in Rupe et al. (2024)."

    The bridge from the fitted DMD error difference ΔKt (Eq. 17) to a claim about dynamical causality is the equivalence ΩC →t ΩE iff ΩC →t_K ΩE. That equivalence is not derived in this paper; the manuscript refers readers to Theorem III.2 of Rupe et al. (2024), whose author list includes two of the present authors (Rupe and DeSantis). No proof, assumption statement, or independent verification of this theorem appears in the manuscript, so the central interpretive claim of Kausal rests on a same-author citation rather than on the derivation chain presented here. This is load-bearing: without the equivalence, Eq. 17 is only a comparison of two predictive models, not a causal measure.

full rationale

Kausal's central novelty is deep-learned observables plus the DMD error-difference causal measure (Eq. 17). The causal measure itself is not fitted to causal labels: ΔKt is evaluated on held-out triplets, and the reported direction and magnitude results are benchmarked against known uncoupled Rössler and reaction-diffusion configurations and against PCMCI+, VARLiNGAM, cLSTM, and TSCI. In that sense the paper's empirical claims are largely self-contained and not circular. However, the step that licenses interpreting ΔKt as 'dynamical causality'—the equivalence ΩC →t ΩE iff ΩC →t_K ΩE—is not derived in this paper; it is referred to Theorem III.2 of Rupe et al. (2024), whose author list includes two present authors (Rupe and DeSantis). This is a load-bearing same-author citation: it bridges a predictive-error comparison to a causal conclusion. The manuscript gives no proof, no statement of assumptions, and no independent verification of this theorem, so the foundational premise is imported rather than established. The alternative reading of Eq. 17 as a generic capacity or predictability comparison (extra inputs in the joint DMD model always reduce in-sample error, and Algorithm 1's shuffled null destroys the effect's autocorrelation) is a statistical-validity concern, not a construction-level circularity; it does not by itself make ΔKt equal to its inputs. Therefore the circularity score reflects the single load-bearing self-citation, with the caveat that the main empirical evaluations are independent and externally grounded.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central method does not introduce physical entities or fit universal constants. Its load-bearing assumptions are borrowed from the Koopman causal framework of Rupe et al. (2024) and the heuristic definition of causal influence as a prediction-error difference. The only extraneous numbers are hyperparameters and protocol choices.

free parameters (3)
  • Dictionary size M (encoder output dimension) = 32 (MLP), 128 (CNN), 500 (RFF)
    Chosen by hand; the paper shows performance improves with M for learned but not for RFF observables (Figure 5).
  • Significance threshold rho_crit = 0.05
    One-sided p-value threshold in Algorithm 1, chosen by hand.
  • Sliding window size and step for AUROC = 10 and 1 (from Runge et al. 2019)
    Protocol choice used to compute causal magnitude and AUROC; not fitted to data.
assumptions (4)
  • domain assumption Local observables associated with each component form closed subspaces of the full RKHS that are preserved by the Koopman operator for the duration of analysis.
    Section 3.2: 'This formulation necessitates that the observables associated with individual components form closed subspaces in the full RKHS'; without this, the marginal/joint comparison is not interpretable as causal.
  • domain assumption The equivalence of dynamical causality and Koopman causality holds (Theorem III.2 of Rupe et al. 2024), and carries over to the finite-rank DMD approximation with learned observables.
    Section 3.2 cites the theorem for the exact operator; the paper does not prove the finite-dimensional analogue.
  • ad hoc to paper The difference in mean-squared prediction error between marginal and joint models is a valid measure of causal influence.
    Eq. (17) defines the causal measure as this difference without formal identification or control for model capacity.
  • domain assumption The dynamical system is autonomous, deterministic, and its phase space can be partitioned into cause, effect, and remainder components.
    Section 2.1 restricts to autonomous systems; Section 3.1 partitions the phase space. Non-stationary or stochastic systems are outside scope.

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Cite this review

Pith. "Pith review of Deep Koopman operator framework for causal discovery in nonlinear dynamical systems." pith.science (2026). https://pith.science/paper/4AGBZPQG

@misc{pith2026250514828,
  author       = {Pith},
  title        = {Pith review of: Deep Koopman operator framework for causal discovery in nonlinear dynamical systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4AGBZPQG}},
  note         = {Machine review of arXiv:2505.14828}
}
read the original abstract

We use a deep Koopman operator-theoretic formalism to develop a novel causal discovery algorithm, Kausal. Causal discovery aims to identify cause-effect mechanisms for better scientific understanding, explainable decision-making, and more accurate modeling. Standard statistical frameworks, such as Granger causality, lack the ability to quantify causal relationships in nonlinear dynamics due to the presence of complex feedback mechanisms, timescale mixing, and nonstationarity. This presents a challenge in studying many real-world systems, such as the Earth's climate. Meanwhile, Koopman operator methods have emerged as a promising tool for approximating nonlinear dynamics in a linear space of observables. In Kausal, we propose to leverage this powerful idea for causal analysis where optimal observables are inferred using deep learning. Causal estimates are then evaluated in a reproducing kernel Hilbert space, and defined as the distance between the marginal dynamics of the effect and the joint dynamics of the cause-effect observables. Our numerical experiments demonstrate Kausal's superior ability in discovering and characterizing causal signals compared to existing approaches of prescribed observables. Lastly, we extend our analysis to observations of El Ni\~no-Southern Oscillation highlighting our algorithm's applicability to real-world phenomena. Our code is available at https://github.com/juannat7/kausal.

Figures

Figures reproduced from arXiv: 2505.14828 by the authors.

Figure 1
Figure 1. Schematic of the Kausal algorithm. We 1) estimate the embeddings using deep learning, and 2) approximate the Koopman operator with dynamic mode decomposition (DMD). Then, 3) we infer causal measures by computing the difference in prediction error between marginal (effect-only) and joint (effect-cause) models. The map from system states to observables is given by a function, ψ : Ω → F. The Koopman operator Kt : F → F… view at source ↗
Figure 2
Figure 2. Illustration of ODEs describing the Rossler Oscillator ¨ system (left), and the corresponding causal graph (right). representing the effect, cause, and remainder components, respectively. Each state ω ∈ Ω is projected into the ef￾fect component using coordinate projections PE, and the evolution of ωE(t) is tracked through the component flow map: PEΦt : ω = [ωE ωC ωR] ⊺ 7→ ωE(t). (12) This framework is flexible enoug… view at source ↗
Figure 4
Figure 4. Conditional forecasts in the (a) true and (b) non-causal direction using MLP kernels. In (a), the addition of ΩC in the joint model improves the forecast of ΩE relative to the marginal model that excludes it. In (b), however, both marginal and joint models make no qualitative difference as ΩE ̸→t K ΩC . 4.2. Reaction-diffusion equation We now showcase Kausal’s scalability to high￾dimensional settings and underscore … view at source ↗
Figures from the paper (5 more)
Figure 6
Figure 6. Figure 6: Causal measure of reaction-diffusion equation by com￾puting the difference between the true causal (∆Kt C,E) and non￾causal direction (∆Kt E,C ) across time shifts t, using an ensemble of randomly initialized CNN kernels (M = 128) to estimate the observables, versus th…
Figure 5
Figure 5. Figure 5: Performance assessment of MLP and RFF kernels for varying dimensionality (i.e. increasing complexity). Kausal is able to extract statistically significant causal signals in the true direction (ΩC →t K ΩE) when compared with the non-causal case (ΩE →t K ΩC ) as t ≫ 0. T…
Figure 8
Figure 8. Figure 8: Causal measure of ENSO for varying nonlinearity strength ϵ using MLP kernels to approximate the observables. Our Kausal framework is able to meaningfully differentiate and characterize nonlinearity in the underlying ENSO dynamics. Kausal is able to track the emergence …
Figure 9
Figure 9. Figure 9: Causal measure of ENSO using real-world data showcasing Kausal’s ability (black line) to capture major El Nino (red shading) ˜ and La Nina (blue shading) events. Events are estimated using NOAA’s Oceanic Ni ˜ no Index ( ˜ Glantz & Ramirez, 2020). The causal measure est…
Figure 10
Figure 10. Figure 10: AUROC scores for the observed ENSO (AUROC > 0.5 shows significant skills better than chance). Overall, Kausal consistently achieves the highest score. The GC-based approaches (PCMCI+, cLSTM and VAR￾LiNGAM) fare worse because their fixed time-lagged em￾beddings cannot …

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.