{"id":"953a85d2-6399-456a-8745-ebbf6ee39d10","arxiv_id":"2505.14828","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A deep Koopman framework called Kausal discovers causal direction and magnitude in nonlinear dynamical systems by comparing joint versus marginal prediction errors in learned observable spaces.","lead":"Kausal uses deep learning to learn Koopman embeddings and then detects causal links between variables by comparing how well models predict an effect when cause information is included versus excluded. On synthetic systems and El Niño data, the learned embeddings find causal direction and magnitude more reliably than fixed observables and standard baselines.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (17) can attribute the joint model's extra input capacity to causation; Algorithm 1's time-shuffle null does not control for this.","rationale":"The paper is a serious empirical contribution: the code is released, the method is tested on multiple nonlinear systems, and the RFF comparisons support the narrower claim that learned observables improve forecasting within the marginal/joint setup. The reader's weakest assumption identifies the same central risk: the causal measure is a conditional-forecast-error gap without an identifiability proof. My stress-test sharpens it: the joint model's advantage can come from extra input dimensionality, a separately trained encoder, and finite-sample chance correlations, and Algorithm 1's permutation null destroys the temporal structure of both series rather than isolating causal coupling. This is load-bearing because every headline result (directional p-values, AUROC, ENSO tracking) is built on Eq. (17). The proposed control experiment would settle whether the asymmetry is causal or merely predictive. Since the paper is already CONDITIONAL on this type of assumption, I do not move the verdict; I would attach the control as a required revision.","tokens_in":19485,"tokens_out":5409,"duration_ms":54412,"concrete_test":"Re-run the coupled Rössler experiment with a control cause: replace ΩC in Eq. (14)-(17) by an independent realization from the same Rössler attractor (or a phase-randomized surrogate of ΩC) that has the same dimension, smoothness, and autocorrelation but no coupling to ΩE, while keeping the MLP architecture and M=32 identical. Apply Algorithm 1 with N=100. If a substantial fraction of such non-causal controls yield ρ<0.05, then Eq. (17) conflates capacity/statistical dependence with causation and the causal-direction claims in Tables 1a/1b need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends on Eq. (17): a lower test error of the joint DMD model relative to the marginal DMD model is interpreted as causal influence. But the joint model differs from the marginal model not only by including ΩC but also by having more input dimensions and a separately learned, higher-capacity encoder. Since K̂ is a least-squares pseudo-inverse (Eq. 8), extra features always reduce in-sample error; on finite test sets they can also reduce error by exploiting chance correlations, shared forcings, or smoothness of the added inputs. Nothing in Eq. (17) separates these mechanisms from true coupling. The significance test in Algorithm 1 does not close the gap: it randomly permutes both ΩC and ΩE, destroying the temporal structure of the effect and the cause, so the null is not 'same system without coupling'. A non-causal control that preserves the autocorrelation and dimensionality of the cause could also produce a significantly positive ∆Kt. Without such a control, the reported asymmetry in Figures 3-6 and the ENSO event tracking in Figure 9 may reflect predictability gains rather than causal influence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19697,"tokens_out":10788,"duration_ms":90987,"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":[{"comment":"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.","section":"Section 3.2, Eq. (17) and Algorithm 1"},{"comment":"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.","section":"Section 2.2 and Appendix C"},{"comment":"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.","section":"Section 4.5 and Appendix D.2"},{"comment":"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.","section":"Section 4.4 and Figure 9"},{"comment":"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.","section":"Section 3.2, Eqs. (17)–(19)"}],"minor_comments":[{"comment":"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.","section":"Algorithm 1"},{"comment":"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.","section":"Table 1b"},{"comment":"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.","section":"Section 4.4"},{"comment":"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.","section":"Eq. (18) and Algorithm 1"},{"comment":"The time step for the reaction-diffusion simulation is given as Δt = 10^3, which is likely a typo for 10^-3; please correct.","section":"Appendix C.2"},{"comment":"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.","section":"Section 4.2, Eq. (21)"},{"comment":"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.","section":"Section 3.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is built directly on a theoretical result from Rupe et al. (2024), one of whose authors (A. Rupe) is a co-author of this paper. This is not a problem per se, but the paper should state the assumptions of that theorem rather than merely citing it, both for self-containedness and to clarify what is being assumed. The real-world ENSO section is honest about being preliminary, but the framing in the abstract overstates what the experiment can show. The paper fits the journal's scope as a machine-learning-for-physics paper, but the causal claims need to be tightened before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new thing here is concrete: Kausal replaces the prescribed random Fourier features in Rupe et al.'s Koopman causal framework with learned neural observables, adds a permutation significance test and ensemble uncertainty, and ships code. That is a real step forward for practical causal analysis in nonlinear, high-dimensional systems, and the paper is written clearly enough that a non-specialist can follow the method.\n\nWhat the paper does well is the empirical demonstration. The controlled Rössler and reaction-diffusion experiments show the method can separate true from false causal direction, and the learned observables clearly beat RFF. The ENSO application tracking major El Niño events is a nice illustration of the kind of downstream use the method enables. I have no reason to doubt the reproducibility: the code is released and the training setup is described in enough detail to re-implement.\n\nThe soft spot is Eq. (17). The causal measure is the test-error gap between a joint model that includes cause observables and a marginal model that does not. But the joint model also has more input dimensions and a separately learned encoder, so the gap can reflect capacity rather than coupling. Algorithm 1's null does not close this gap: it randomly permutes both cause and effect, destroying temporal structure and autocorrelation, so it is not a 'same system without coupling' control. A non-causal input with matched dimensionality and autocorrelation could easily produce the same asymmetry. The paper should add such a control, or at least acknowledge that the measure is a heuristic. This is a moderate flaw, not a fatal one—the simulations with known ground truth support the method's practical value, but the theoretical identification claim is not established.\n\nMinor issues: the loss for training the neural observables is not fully specified in the main text (appendix gives architecture but not the exact objective), baseline comparisons on reaction-diffusion use a spatially averaged time series for the baselines while Kausal gets the full grid, and the AUROC comparisons lack error bars. These are all fixable.\n\nBottom line: this paper is for Earth scientists and applied dynamical systems researchers who want a practical tool, not for someone seeking rigorous causal identification guarantees. It deserves a serious referee, but the authors should add a proper non-causal control and tone down the causal language.","headline":"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.","tokens_in":724,"tokens_out":866,"would_cite":false,"duration_ms":30261,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Kausal combines deep Koopman operators with causal discovery: learned observables make the causal signal between cause and effect measurable in nonlinear dynamical systems.","keywords":["causal discovery","Koopman operator","deep learning","dynamical systems","ENSO","dynamic mode decomposition","reproducing kernel Hilbert space","nonlinear dynamics"],"falsifier":"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.","tokens_in":19282,"feed_emoji":"🌊","tokens_out":5731,"duration_ms":51275,"temperature":0.7,"pith_summary":"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.","feed_headline":"Learned Koopman operators detect true causes in nonlinear systems","feed_subtitle":"Kausal learns observables with neural networks, then reads causality from the joint-model forecast gain over the effect-only model.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the theoretical equivalence between dynamical causality and Koopman causality that Kausal's measure builds on.","marker":"Rupe et al. (2024)"},{"why":"introduces the operator-theoretic view of nonlinear dynamics as linear evolution of observables.","marker":"Koopman (1931)"},{"why":"provides the identity-observable augmentation and finite linear representation of nonlinear dynamics used in the marginal and joint models.","marker":"Brunton et al. (2016)"},{"why":"supplies the random Fourier feature dictionaries that serve as the prescribed-observable baseline.","marker":"Rahimi & Recht (2007)"},{"why":"establishes convergence of the finite-rank DMD approximation to the Koopman operator as data and dictionary size grow.","marker":"Korda & Mezić (2018)"},{"why":"underpins the use of deep learning to learn linearizing embeddings for nonlinear dynamics.","marker":"Lusch et al. (2018)"},{"why":"defines the conceptual ENSO model whose causal structure Kausal is tested against.","marker":"Jin (1997)"},{"why":"provides the ECMWF ocean reanalysis data used in the real-world ENSO experiment.","marker":"Zuo et al. (2019)"}],"fun_headline_variants":["Deep Koopman operators learn causal structure in nonlinear dynamics","Kausal: neural network learns Koopman observables for causal discovery","Neural Koopman framework identifies true causes in chaotic systems","Deep learning plus Koopman theory reveals cause-effect in dynamics","Machine-learned observables sharpen causal detection in nonlinear systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Deep Koopman operators learn causal structure in nonlinear dynamics","Kausal: neural network learns Koopman observables for causal discovery","Neural Koopman framework identifies true causes in chaotic systems","Deep learning plus Koopman theory reveals cause-effect in dynamics","Machine-learned observables sharpen causal detection in nonlinear systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000607,"raw_usage":{"total_tokens":2855,"prompt_tokens":999,"completion_tokens":1856,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":1770}},"tokens_in":615,"tokens_out":1856,"duration_ms":12765,"temperature":1.0,"reasoning_tokens":1770,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:28:56.147376+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the operator-theoretic view of nonlinear dynamics as linear evolution of observables."},{"cited_title":"and Recht, B","cited_arxiv_id":null,"evidence_quote":"supplies the random Fourier feature dictionaries that serve as the prescribed-observable baseline."},{"cited_title":"A., Tietsche, S., Mogensen, K., and Mayer, M","cited_arxiv_id":null,"evidence_quote":"provides the ECMWF ocean reanalysis data used in the real-world ENSO experiment."}],"review_version":1}