{"id":"74252571-fe51-4ac8-a8dc-d7b13443178c","arxiv_id":"2508.08247","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A quadratic classifier logit built from learned state embeddings converts forward-versus-reversed trajectory discrimination into a scalable entropy-production estimate plus a low-dimensional map of irreversible flow.","lead":"Researchers introduce LENS, a machine-learning tool that reads long time series from steady nonequilibrium systems and learns a low-dimensional set of coordinates where energy dissipation actually happens. LENS also estimates the total entropy production rate, and tests on model systems match exact calculations to within a few percent.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Identification claim is unsupported for nonlinear systems: the quadratic logit is representationally non-unique, and the paper defers the needed characterization; EPR estimation alone does not establish that learned embeddings expose latent irreversible coordinates.","rationale":"The reader's conditional verdict already targets the same gap, and my analysis sharpens it: the non-uniqueness of φ is not merely a missing theorem but follows from the exact invariance of the logit under invertible linear transformations of the representation; the paper's own Discussion and App. A4 concede that the nonlinear identification question is open. The synthetic test would settle it directly. If the test fails, the headline claim should be downgraded to 'low-dimensional EPR estimate' rather than 'identified irreversible degrees of freedom'; if it passes, the claim is supported. The EPR estimation component is solid and well benchmarked, and the code is public, so REJECT would be too strong; CONDITIONAL remains the right verdict, exactly as the reader proposed. Hence no change to the verdict.","tokens_in":34314,"tokens_out":26197,"duration_ms":253614,"concrete_test":"Build a synthetic 'nonlinear latent linear' benchmark: let z_t ∈ R^2 follow a linear OU process with known force matrix Λ, and observe x_t = h(z_t) where h is an invertible nonlinear map (e.g., h(z) = z + ε sin(ω z) composed with a fixed random invertible MLP). Train LENS with nonlinear φ and M = 2 using the released code, then compute the canonical correlations between the learned φ(x) and the true latent z over test data, after allowing the gauge (permutations/rotations within the 2×2 block), and repeat over 5 seeds. If the best canonical correlation is not close to 1 while EPR is accurately estimated, the identification claim fails; as a control, permute the two latent coordinates and re-run, and check that the learned representation permutes correspondingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest load-bearing premise is that the learned nonlinear embedding φ identifies the latent irreversible degrees of freedom, not merely that the average logit estimates the scalar EPR. The universality theorem (App. B3) proves that for any smooth antisymmetric logit there exists some φ, A, B approximating it; it does not constrain which φ is found. For any invertible linear T, the logit (3) is unchanged under φ → Tφ, A → T^{-T}AT^{-1}, B → T^{-T}BT^{-1}, so the objective cannot select a physically unique coordinate system. The block-diagonal form of A (Cor. 1.1) is a canonical gauge for A, not an identifiability result for φ; B transforms simultaneously in an unconstrained way. The only case with a precise coordinate interpretation is linear dynamics with linear representations (App. F), where LENS learns a low-rank approximation to the force matrix Λ. For the motivating class named in the Discussion—nonlinear observations generated by latent linear irreversible dynamics—the paper explicitly leaves 'a mathematical and empirical characterization' to future work, and App. A4 says consistency of φ in nonlinear systems is open. The periodic-potential experiment does not close this gap: the periodic coordinate is known in advance and is inserted by hand as a sine layer (App. A5), so the representation does not discover an unknown latent coordinate. If φ is merely a flexible basis that fits the forward/reverse contrast, the embedding is arbitrary while the EPR estimate remains accurate, reducing LENS to NEEP-style scalar estimation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces LENS, a contrastive learning method that parameterizes the forward/reverse classifier logit as a quadratic form of a learned state representation. The average logit estimates the entropy production rate (EPR), while the learned representation is intended to expose the system's irreversible degrees of freedom. The authors prove a universality result for the quadratic logit, derive an exact low-rank interpretation for linear dynamics, and test the method on three systems: coupled harmonic beads, a driven particle in a periodic potential, and the two-dimensional complex Ginzburg-Landau model. Code is provided.","tokens_in":34691,"tokens_out":4380,"duration_ms":49183,"significance":"If the identification claim holds, LENS would be a valuable advance over existing EPR-only estimators because it adds interpretable coordinates to the entropy production estimate. The paper has real strengths: the analytic EPR benchmarks for the N-bead and periodic-potential systems are convincing, the low-rank theory for linear systems in App. F gives a parameter-free prediction that matches the observed scaling with representation dimension, error bars over random seeds are reported, and the code is publicly available. However, the central identification claim is currently established only for linear dynamics. For nonlinear systems the evidence consists of visual flow inspection, a known periodic coordinate inserted by hand as a sine layer, and a deterministic CGL example with no ground-truth latent coordinates; the Discussion and App. A4 explicitly defer the needed characterization. As written, the paper is a solid EPR estimator with a promising but unproven coordinate-discovery component.","major_comments":[{"comment":"The paper's central claim is that LENS identifies latent irreversible degrees of freedom in nonlinear systems, but the nonlinear experiments either use a known periodic coordinate inserted by hand (Fig. 3 and App. A5) or are interpreted through visual flow inspection without ground truth (Fig. 4). The Discussion states that 'a mathematical and empirical characterization of LENS in such scenarios is an interesting direction for future work,' and App. A4 says that consistency of phi in nonlinear systems is open. These statements concede exactly the load-bearing part of the identification claim. I request either (i) a benchmark with known latent irreversible coordinates, for example nonlinear observations generated by latent linear dynamics as named in the Discussion, in which the learned phi is compared quantitatively with the ground-truth coordinates (e.g., by canonical-correlation or shared-variance measures), or (ii) a substantial tempering of the identification claim in the title and abstract to reflect that coordinate identification is currently demonstrated for linear systems only.","section":"Discussion and Outlook; App. A4"},{"comment":"The logit parameterization is gauge-invariant under phi -> T phi, A -> T^{-T} A T^{-1}, B -> T^{-T} B T^{-1} for any invertible T, so the LENS objective does not select a unique embedding. The Schur block-diagonal form of A (Cor. 1.1) canonicalizes A alone; because B transforms simultaneously and is otherwise unconstrained, the combined gauge is not fixed. App. A4 in fact acknowledges that 'it is not possible to simultaneously block-diagonalize the skew-symmetric matrix A and diagonalize the symmetric matrix B.' Thus the learned coordinates are not identifiable as physical degrees of freedom without additional assumptions. At minimum, the paper should state the equivalence class of representations that the objective identifies and prove or test which member of that class is selected by the optimizer; the eigenvalue reproducibility claim in Fig. S6 concerns only the directly-learned linear projection and does not cover nonlinear phi.","section":"Eq. (3); App. B3; App. A4"},{"comment":"The driven-periodic-potential experiment does not provide evidence for discovery of an unknown latent coordinate. Because the periodicity is known a priori, a sine activation layer is inserted after the first dense layer and is described as 'necessary for stable maximization' in App. A5. The learned representation is therefore strongly biased toward the known periodic variable rather than discovering it from data. This leaves the EPR accuracy results intact, but it removes this experiment as support for the identification claim.","section":"Fig. 3; App. A5"}],"minor_comments":[{"comment":"After Eq. (S44), the text reads 'This implies that A^T = A, so A is skew-symmetric as required'; it should read A^T = -A.","section":"App. B3, proof of Lemma 2"},{"comment":"The main text refers to 'four different temperature profiles' but does not name them at first mention; the profiles are only listed in the Fig. 2 caption. Please cite Fig. 2(a) at that point so the reader can identify the profiles.","section":"Fig. 2(a) and main text after Eq. (4)"},{"comment":"Table S6 lists temperature ratios {0.05, 0.1, 0.25, ..., 0.9999} that appear copied from the N=2 bead system, while the surrounding text describes a sample-efficiency comparison for the N=16 alternate-profile system with a single temperature ratio of 0.1. Please correct the table or clarify which ratios were used for the N=16 experiment.","section":"Table S6 and App. A3b"},{"comment":"Because the CGL dynamics are deterministic, the text correctly notes that the EPR estimates diverge logarithmically; the y-axis label 'EPR Estimate' may nonetheless mislead readers. Consider labeling this quantity as a finite-time irreversibility proxy and specifying the observation time window used.","section":"Fig. 4(a)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the journal's scope and the EPR-estimation results are solid. The main risk is that the title and abstract promise identification of nonequilibrium degrees of freedom, while the identification claim for nonlinear systems is explicitly deferred. If the authors add a nonlinear latent-coordinate benchmark and clarify the identifiability statement, I would expect the paper to be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a well-executed paper with a real new result in the linear case, and a headline claim about nonlinear systems that the authors themselves flag as unfinished. Worth sending to a serious referee, but the referee should push them to either prove or severely qualify the identification claim.\n\nWhat's actually new: the quadratic-logit parameterization S = phi(x')^T A phi(x) + 1/2 phi(x')^T B phi(x') - 1/2 phi(x)^T B phi(x), with A block-diagonal skew-symmetric and B symmetric. The universality theorem (App. B3) is clean and correct as far as I can see: any smooth antisymmetric logit can be approximated this way. The more interesting result is in App. F: for linear dynamics with linear representations, the learned matrices converge to a rank-M approximation of the thermodynamic force matrix Lambda = D^{-1}G + C^{-1}. That gives a concrete physical meaning to the embeddings, and the numerical experiments on the N-bead system match the reduced theory to within error bars. The comparison to PCA is fair, and the code is available. The objective equivalence to NEEP (App. E) is openly acknowledged, which I appreciate.\n\nSoft spots, in order of severity. First, for nonlinear systems the identification claim is unsupported. The logit is invariant under phi -> T phi, A -> T^{-T} A T^{-1}, B -> T^{-T} B T^{-1}, so the objective alone cannot select a physically unique coordinate system. The block-diagonal constraint on A fixes part of the gauge but not B, and no argument is given that the learned phi aligns with latent irreversible coordinates. The authors know this: the Discussion explicitly defers a characterization of LENS for nonlinear latent linear dynamics, and App. A4 says consistency of phi in nonlinear systems is open. The periodic potential experiment does not close the gap, because the periodic coordinate is known ahead of time and is inserted via a sine activation layer. Second, the CGL application is heuristic: the dynamics are deterministic, EPR diverges, and the claimed transition detection is based on visual flow inspection. Third, minor: the EPR positivity floor of 1e-8 is hand-set, but it only matters at tiny EPR and is disclosed.\n\nNone of these are fatal. The EPR estimation component is solid and comparable to NEEP, and the linear-system identifiability result is new and valuable. The paper would benefit from either an explicit identifiability analysis for a tractable nonlinear class (e.g., latent linear dynamics with nonlinear observation maps), or a revised title and abstract that restrict the identification claim to the linear setting. I'd send it to peer review.","headline":"A solid method paper for linear systems whose nonlinear identification claim rests on an unproven identifiability assumption.","tokens_in":35164,"tokens_out":2604,"would_cite":true,"duration_ms":27534,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["05.70.Ln","05.40.-a","05.10.Gg"],"model":"deepseek-v4-flash","headline":"A neural network whose logit is a quadratic form of learned embeddings isolates the few coordinates that actually produce entropy, and estimates the entropy production rate from raw time series at the same time.","keywords":["entropy production","nonequilibrium steady state","irreversible degrees of freedom","representation learning","contrastive learning","low-rank approximation","stochastic dynamics","dimensionality reduction"],"falsifier":"A concrete falsifier would be a simulation of a nonlinear stochastic system with known latent linear irreversible coordinates (e.g., two latent modes undergoing a circulating flow, observed through a highly nonlinear invertible map). If LENS, trained with $M=2$, yields embeddings that are not related to the true latent coordinates by any smooth invertible transformation, or if the estimated EPR from the learned representation fails to reproduce the analytic EPR despite the classifier achieving low validation loss, then the identification claim would be refuted. Another check: for a system where the true irreversible coordinates are known a priori, the eigenvalues of the learned $A_\\theta$ should match the true circulation strengths up to permutation; if they do not, the learned representation is not uncovering the irreversible degrees of freedom.","tokens_in":34122,"feed_emoji":"🔥","tokens_out":2031,"duration_ms":23565,"temperature":0.7,"pith_summary":"LENS (Latent Embeddings of Nonequilibrium Systems) is a model-free method that learns a low-dimensional representation of a high-dimensional stochastic system in a nonequilibrium steady state, such that the representation exposes the degrees of freedom that generate entropy production. The method trains a classifier to distinguish forward from time-reversed transitions, but instead of using a free-form logit, it constrains the logit to a specific quadratic form built on learned embeddings. This constraint makes the learned representation interpretable as the irreversible flow coordinates, and the classifier's average logit directly estimates the entropy production rate. The paper demonstrates on simulated linear and nonlinear systems that LENS both identifies circulating flows in the learned representation space and estimates entropy production rates with small error, outperforming PCA-based projections.","feed_headline":"Neural net finds the coordinates that waste energy","feed_subtitle":"A new representation-learning method, LENS, isolates entropy-producing directions in high-dimensional stochastic data.","key_machinery":"The central object is the LENS logit $S_\\theta(\\vec x,\\vec x')$, a quadratic form on the learned representation $\\vec\\phi_\\theta(\\vec x)$, antisymmetric under exchange of its arguments by construction. The structure combines an antisymmetric bilinear term $\\vec\\phi^T A \\vec\\phi$ (which captures the circulating, entropy-producing flow) with a difference of symmetric quadratic forms (which captures the potential-like, reversible part). This parametrization is universal for approximating any antisymmetric logit (Theorem 1 and Appendix B3) and yields a precise low-rank geometric interpretation for linear systems: maximizing the LENS objective leads to learning $\\tilde K + \\tilde L = P^T(K+L)P \\approx \\Lambda = D^{-1}G + C^{-1}$, the thermodynamic force matrix. The EPR is then the expectation of the logit, $\\dot S = \\mathbb E[S_\\theta(\\vec x,\\vec x')]$, which is directly computable from the classifier output.","core_discovery":"The central claim is that for a Markovian nonequilibrium steady state, the optimal classifier logit between forward and reverse transition probabilities, $S^*(\\vec x,\\vec x') = \\log[p(\\vec x\\to\\vec x')/p(\\vec x'\\to\\vec x)]$, can be parametrized without loss of expressiveness as $S_\\theta(\\vec x,\\vec x') = \\vec\\phi_\\theta(\\vec x')^T A_\\theta \\vec\\phi_\\theta(\\vec x) + \\tfrac12 \\vec\\phi_\\theta(\\vec x')^T B_\\theta \\vec\\phi_\\theta(\\vec x') - \\tfrac12 \\vec\\phi_\\theta(\\vec x)^T B_\\theta \\vec\\phi_\\theta(\\vec x)$, where $\\vec\\phi_\\theta$ is a learned nonlinear representation, $A_\\theta$ is block-diagonal skew-symmetric, and $B_\\theta$ is symmetric. With this parametrization, the learned representations correspond to the irreversible degrees of freedom: for linear systems, LENS learns a low-rank approximation to the thermodynamic force matrix $\\Lambda = D^{-1}G + C^{-1}$, so the entropy production rate from the low-dimensional projection converges monotonically to the true EPR as the representation dimension $M$ increases. For nonlinear systems, the quadratic form is universal (proved via Stone-Weierstrass) in approximating any smooth antisymmetric logit, so the same construction extracts entropy-producing coordinates without requiring knowledge of the dynamics.","pith_inferences":["Editorial inference: If LENS embeddings indeed align with the true latent irreversible coordinates in nonlinear systems, the method could be used not only to estimate EPR but also to construct reduced models of the nonequilibrium dynamics, by projecting the learned representation onto the dissipative subspace and integrating that low-dimensional flow.","Editorial inference: The universality proof applies to any smooth antisymmetric logit, suggesting that the same quadratic-form parametrization could serve as a universal approximator for other antisymmetric functions arising in stochastic thermodynamics, such as time-asymmetric correlation functions or the full counting statistics generator.","Editorial inference: A testable extension is to apply LENS to systems with explicit latent linear irreversible dynamics and a nonlinear observation map (e.g., a chaotic measurement function), to empirically check whether the learned embedding recovers the latent coordinates up to an invertible transformation; the paper states this characterization is future work.","Editorial inference: The method's dependence on a binary cross-entropy objective (rather than the f-divergence used by NEEP) appears to provide training stability at high EPR, so a hybrid objective that interpolates between BCE and f-divergence could potentially improve accuracy at intermediate data sizes."],"forward_implications":["For linear stochastic systems with unknown parameters, LENS provides a scalable estimator of the entropy production rate that requires no knowledge of the drift or diffusion matrices, and the learned projection $P$ reveals which combinations of coordinates carry the irreversible driving.","For nonlinear systems with latent linear irreversible dynamics, the universality result suggests that LENS can discover the latent coordinates that generate entropy, even when the observed high-dimensional dynamics are nonlinear functions of those coordinates.","The method is applicable to partially observed data (e.g., rendered images of bead positions, video frames of complex fields) by using convolutional encoders, making it a candidate for estimating dissipation in experimental active matter and biological systems from raw imaging time series.","The low-rank interpretation gives a principled way to choose the representation dimension $M$: the relative EPR error decreases monotonically with $M$ and vanishes when $M = N$, providing a diagnostic for how many irreversible degrees of freedom are needed.","By identifying which state-space directions carry the entropy production, LENS can serve as a dimensionality-reduction tool that preserves thermodynamic relevance rather than mere reconstruction, complementing or replacing PCA in nonequilibrium studies."],"supporting_citations":[{"why":"Introduces NEEP, the neural estimator of entropy production via f-divergence minimization; LENS is contrasted with it and shown to be equivalent in the infinitesimal-time limit.","marker":"[18]"},{"why":"Shows that low-dimensional projections underestimate entropy production, motivating the low-rank approximation interpretation and the PCA baseline comparison.","marker":"[17]"},{"why":"Provides the definition of entropy production rate as KL divergence between forward and reverse path distributions, which is the foundation of the LENS objective.","marker":"[25]"},{"why":"Supplies the phase diagram of the two-dimensional complex Ginzburg-Landau equation used to locate the defect turbulence transition that LENS is tested on.","marker":"[31]"},{"why":"Gives the analytic solution for the entropy production rate of an overdamped particle in a periodic potential, which is the exact baseline for the nonlinear test system.","marker":"[30]"},{"why":"Provides the coupled bead-spring model with local reservoirs, used as the linear test system with analytic EPR.","marker":"[7]"}],"fun_headline_variants":["Neural net identifies entropy-producing coordinates","LENS: learning irreversible degrees of freedom","Model-free AI finds hidden entropy sources","Learning embeddings that capture entropy production","New method reveals waste in high-dimensional data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's identification claim relies on the assumption that, for nonlinear stochastic systems, optimizing the quadratic logit with a learned representation will not merely fit the scalar entropy production but will actually align the learned embeddings with the latent irreversible degrees of freedom; the authors prove universality for approximating any antisymmetric logit and give a precise low-rank interpretation only for linear dynamics, while explicitly stating that a characterization for nonlinear latent linear dynamics is future work.","fun_headline_variants_meta":{"raw":{"variants":["Neural net identifies entropy-producing coordinates","LENS: learning irreversible degrees of freedom","Model-free AI finds hidden entropy sources","Learning embeddings that capture entropy production","New method reveals waste in high-dimensional data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000217,"raw_usage":{"total_tokens":1451,"prompt_tokens":977,"completion_tokens":474,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":593,"completion_tokens_details":{"reasoning_tokens":412}},"tokens_in":593,"tokens_out":474,"duration_ms":5037,"temperature":1.0,"reasoning_tokens":412,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:36:33.341901+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete falsifier would be a simulation of a nonlinear stochastic system with known latent linear irreversible coordinates (e.g., two latent modes undergoing a circulating flow, observed through a highly nonlinear invertible map). If LENS, trained with $M=2$, yields embeddings that are not related to the true latent coordinates by any smooth invertible transformation, or if the estimated EPR from the learned representation fails to reproduce the analytic EPR despite the classifier achieving low validation loss, then the identification claim would be refuted. Another check: for a system where the true irreversible coordinates are known a priori, the eigenvalues of the learned $A_\\theta$ should match the true circulation strengths up to permutation; if they do not, the learned representation is not uncovering the irreversible degrees of freedom.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces NEEP, the neural estimator of entropy production via f-divergence minimization; LENS is contrasted with it and shown to be equivalent in the infinitesimal-time limit."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that low-dimensional projections underestimate entropy production, motivating the low-rank approximation interpretation and the PCA baseline comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the phase diagram of the two-dimensional complex Ginzburg-Landau equation used to locate the defect turbulence transition that LENS is tested on."},{"cited_title":"Bae, D.-K","cited_arxiv_id":null,"evidence_quote":"Gives the analytic solution for the entropy production rate of an overdamped particle in a periodic potential, which is the exact baseline for the nonlinear test system."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the coupled bead-spring model with local reservoirs, used as the linear test system with analytic EPR."}],"review_version":1}