{"id":"8433d7f1-7539-4f9b-afd1-daee766df883","arxiv_id":"2501.17993","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A machine learning framework learns thermodynamically consistent internal variables and Markovian evolution equations directly from stochastic particle trajectories, validated on an analytically solvable trap model and a phase-transforming chain.","lead":"This paper combines information bottleneck encoders, normalizing flows, and thermodynamically constrained neural networks to discover internal state variables and their evolution equations from microscopic particle simulations. The method is tested on a trapped colloidal particle and a phase-transforming chain, reproducing microscopic statistics and dynamics on held-out protocols.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The sufficiency of the learned internal variables for Markovianity is not tested: the same-time IB target and marginal-only decoder validation leave the central no-memory claim unsupported.","rationale":"The reader's conditional verdict is appropriate. The two examples provide genuine evidence: Example 1 is analytically solvable and the method recovers the free energy and dissipation potentials, with the learned internal variable collapsing to a bijection of the mean displacement; Example 2 shows held-out predictive errors of a few percent for both GMM and CNF decoders, and CNFs outperform GMMs. Neither of these facts, however, discharges the sufficiency assumption. The central concern is not that a finite-dimensional Markovian embedding cannot exist for these systems, but that the training procedure does not select for it and the validation does not measure it. Setting X=Y means the IB objective is, in effect, an autoencoder; only the VONNs dynamics term penalizes non-Markovianity, and that penalty can be minimized by an over-flexible Markovian surrogate. The marginal-only decoder in Example 2 makes the gap larger, because correlations between springs are part of the microscopic distribution that the theory claims to determine. None of this warrants rejection: the paper is careful, the experiments are well designed, and the methodology is a plausible step toward automated internal-variable discovery. But the central claim is broader than what is verified, so the correct verdict remains conditional. I agree partially with the reader: the same-time X=Y issue is identified in the reader's weakest_assumption, but the marginal-only validation is a distinct and equally load-bearing gap.","tokens_in":23863,"tokens_out":6033,"duration_ms":71819,"concrete_test":"Use the Example 2 Langevin trajectories to compute the conditional mutual information CMI = I(ε_i(t); ε_{i+1}(t) | z_i(t), z_{i+1}(t)) with a k-nearest-neighbor estimator at several times t and springs i, comparing against a bootstrap null from shuffled realizations. If CMI is significantly above zero, then the learned state variables do not determine the joint microstate distribution, directly contradicting the sufficiency assumption and undermining the Markovianity claim. As a complementary check, retrain with a time-lagged target Y=x(t+τ) in Eq. (2) and compare the learned α and held-out errors; a material change would indicate that the same-time X=Y choice misses future-predictive information.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weakness is that the method never independently verifies the sufficiency condition on which Markovianity rests. In Section 2.2, the IB target is set to the same-time microstate (X=Y=x(t)) and the term I(Z;X) is dropped, so the encoder is trained only to reconstruct the current conditional distribution, not to be predictive of the future. The VONNs losses in Eqs. (14)-(15) then fit a Markovian deterministic evolution from z(t) to z(t+Δt); but any sufficiently flexible Markovian ansatz can fit finite training data even when the true coarse-grained dynamics retains memory, so agreement tells us only that the ansatz is expressive, not that z is sufficient. In Example 2 the decoder does not even represent the full microstate: q(ε_i|z_i) is a one-dimensional conditional model for each spring and the loss in Eq. (51) sums marginal log-likelihoods, so joint correlations among strains are not characterized. Figures 16 and 20 validate only marginals. If adjacent strains are correlated given z, the central assumption that z determines the microscopic probability distribution is violated, and the 'no memory effects' conclusion is unsupported. The held-out errors are encouraging but only cover a few protocols and do not probe whether the learned variables would remain sufficient outside this training family.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes IB-VONNs, a framework that combines an information-bottleneck encoder--decoder with conditional normalizing flows and Variational Onsager Neural Networks to discover internal variables and thermodynamically consistent, Markovian evolution equations from microscopic stochastic dynamics. The encoder maps ensembles of microstates to permutation-invariant macroscopic state variables, the decoder reconstructs the conditional microscopic distribution, and VONNs learn free-energy and dual-dissipation potentials whose gradients drive the macroscopic dynamics. The method is tested on two overdamped Langevin systems: a single colloidal particle in an optical trap with an analytic Gaussian solution, and a one-dimensional double-well mass-spring chain with multimodal distributions, using both Gaussian-mixture and conditional-normalizing-flow decoders. Reported held-out errors are small for both state variables and external force, and predictions generalize to a sinusoidal protocol not used in training.","tokens_in":24164,"tokens_out":4713,"duration_ms":53651,"significance":"If the central claim is established, the paper offers a genuinely useful route from stochastic microstate data to thermodynamically consistent internal-variable models, a long-standing gap in non-equilibrium statistical mechanics and continuum mechanics. The methodology is principled: the set-invariant encoder respects statistical-mechanical indistinguishability of realizations, CNFs remove the need for a hand-chosen distribution family, and VONNs enforce thermodynamic consistency in the learned potentials. The use of an analytically solvable benchmark, held-out data, and an unseen protocol are strengths, as is the explicit reporting of quantitative errors. The main limitation is that the paper's two load-bearing assumptions -- that the learned state variables are sufficient for the full microscopic distribution and that the resulting coarse-grained dynamics are Markovian -- are not independently validated, only encoded in the model architecture and checked indirectly through predictive accuracy on the training family.","major_comments":[{"comment":"The IB target is set to the same-time microstate (X = Y) and the complexity term I(Z;X) is dropped, so the encoder is not explicitly trained to be predictive of the future. The VONN dynamics loss then fits a Markovian evolution from z(t) to z(t+Δt) using only current state variables. Because a sufficiently expressive Markovian ansatz can fit finite training data even when the true coarse-grained dynamics retain memory, the small training and test errors do not by themselves establish the paper's no-memory claim. I recommend adding explicit diagnostics for Markovianity, for example testing conditional independence p(z_{t+1}|z_t,z_{t-1}) = p(z_{t+1}|z_t) on held-out trajectories, or comparing the IB-VONN model against a history-dependent baseline with the same encoder and reporting the difference in generalization.","section":"Section 2.2, Eq. (2) and Section 2.4, Eqs. (14)--(15)"},{"comment":"The decoder in Example 2 is a product of per-spring one-dimensional conditional models: the loss in Eq. (51) sums marginal log-likelihoods log q(ε_i|z_i) over springs, and Figures 16 and 20 validate only the marginal distributions. This does not establish that the state variables z characterize the microscopic probability distribution, which is one of the paper's central assumptions stated in Section 2. If the spring strains are correlated conditional on z, the learned internal variables may be sufficient for the marginals but not for the joint distribution, and the derived Markovian evolution would not be a consequence of the assumed sufficiency. The authors should either measure joint conditional correlations (e.g., compare residual covariance of pairs of strains given z) or replace the marginal decoder with a multivariate conditional model and report joint likelihoods.","section":"Example 2, Eq. (51) and Figures 16, 20"},{"comment":"The conclusions state that the state variables 'lead to Markovian dynamics' and that the framework can 'bridge statistical mechanics and thermodynamics' away from equilibrium. In the current paper this is a property of the model ansatz rather than a validated property of the learned representation. The theoretical link [28] requires the conditional distribution of the microstate given z to coincide with the true microscopic distribution, a condition that is assumed rather than tested. I recommend softening the claim to 'the learned dynamics are consistent with a Markovian, thermodynamically admissible representation' until the sufficiency of the latent state is directly verified or the assumption is stated as an explicit, justified approximation.","section":"Section 5 and abstract"}],"minor_comments":[{"comment":"The word 'famework' should be 'framework' in the sentence 'we also test the IB-VONNs famework on data generated by a smoothed linear pulling protocol.'","section":"Section 3.3, last paragraph"},{"comment":"There are typos: 'miscroscopic' should be 'microscopic' in the first paragraph of Section 4.4, and 'mean stain' in the Figure 13 caption should be 'mean strain'.","section":"Section 4.4, first paragraph and Figure 13 caption"},{"comment":"The statement that the encoder is 'maximally predictive of the future state' is not justified by the IB formulation in this paper, because the target Y is the current microstate, not a future quantity; rephrasing this as a design goal rather than a property of the loss function would avoid overstating the role of the IB term.","section":"Section 2.2, paragraph after Eq. (4)"},{"comment":"The GMM decoder uses the fixed peak-distance ε_h, which is a physics-informed assumption; this is worth stating more prominently as a modeling choice that is later removed by the CNF version, so that readers do not interpret it as part of the general IB-VONNs method.","section":"Section 4.3, Eq. (54)"},{"comment":"The loss weights λ_pdf, λ_ε, and λ_α are hand-tuned, and the conclusion acknowledges this as an open issue; a brief discussion of sensitivity to these weights, or at least the observed ranges over which the reported errors are stable, would increase the reproducibility of the numerical results.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of the journal and cites appropriate prior work; the main risk is that the sufficiency/no-memory assumption is under-tested, so the revision should focus on adding direct validation of the Markovian property and the joint conditional distribution rather than on polishing the presentation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper is a genuine methodological contribution and the experiments are careful, but the central claim—that the learned internal variables are sufficient to eliminate memory effects—is not actually tested. What the paper does well: it builds a framework that combines a permutation-invariant encoder (Deep Sets), a conditional normalizing flow decoder, and VONNs to learn internal variables and thermodynamically consistent evolution equations from stochastic many-body trajectory data. Example 1 is a nice validation: the optical trap is analytically solvable, and the method recovers the free energy and dissipation potentials with moderate errors and predicts the external force (0.17% L2) and internal variable (2.7% L2) on held-out pulling velocities, plus a different protocol. Example 2 is more demanding—a mass-spring chain with double-well potential at high temperature—and the method works with both GMM and CNF decoders, with held-out errors in the 1-6% range and better generalization on a sinusoidal protocol with CNFs. That is real evidence the framework is not just curve fitting.\n\nWhere it falls short: the paper does not verify the sufficiency condition on which Markovianity rests. In Section 2.2 they set X=Y (same-time microstate) and drop the complexity term I(Z;X), so the encoder learns to reconstruct the current distribution, not to predict the future. The VONNs loss then fits a Markovian map from z(t) to z(t+dt); any sufficiently expressive Markovian model can fit finite training data even when the true coarse-grained dynamics has memory. Good held-out error is encouraging but does not establish that z is sufficient. In Example 2 the decoder only models one-dimensional marginals, q(ε_i|z_i), and the loss sums marginal log-likelihoods; joint correlations among springs are never checked. Figures 16 and 20 validate only marginals. If adjacent strains are correlated given z, the central 'no memory effects' conclusion is unsupported. There are also smaller issues: loss weights and the number of internal variables are hand-tuned, the method is called 'information bottleneck' even though the bottleneck term is dropped (the paper says this), and no code is released. These are addressable.\n\nWho should read it: people working on multiscale modeling, internal variables, and machine learning for nonequilibrium thermodynamics. It deserves a serious referee. I would ask for a direct test of sufficiency—for instance, a conditional independence check of the next microstate given z and history, or comparison against a non-Markovian baseline—and for validation of joint distributions, not just marginals. Recommendation: send to peer review; the framework is worth engaging with, but the central claim needs to be tested explicitly.","headline":"A solid ML framework for learning internal variables with thermodynamic consistency, but the no-memory claim is backed by predictive accuracy, not a direct sufficiency test.","tokens_in":24652,"tokens_out":3601,"would_cite":true,"duration_ms":33073,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82C31","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes a machine-learning framework, IB-VONNs, aimed at automatically discovering internal variables and thermodynamically consistent evolution equations from stochastic microscopic particle data.","keywords":["non-equilibrium thermodynamics","internal variables","information bottleneck","conditional normalizing flows","Variational Onsager Neural Networks","overdamped Langevin dynamics","coarse-graining","machine learning"],"falsifier":"Measure the residual mutual information between the future microscopic state and the past history of the state variables, conditioned on the current $(\\chi,\\alpha)$; if this residual fails to vanish for any finite-dimensional internal variable set, the central Markovian sufficiency premise is false.","tokens_in":23644,"feed_emoji":"⚛️","tokens_out":4882,"duration_ms":49202,"temperature":0.7,"pith_summary":"This paper proposes a machine-learning strategy, IB-VONNs, aimed at bridging statistical mechanics and thermodynamics away from equilibrium. The claim is that, from stochastic microscopic data, the framework can automatically discover macroscopic internal variables that are functions of the microstate, recover the microscopic probability distribution conditioned on those variables, and learn Markovian evolution equations that respect the second law. If correct, continuum models of inelastic materials would no longer rely on phenomenologically chosen internal variables, and physics-based structure-property relations could be derived directly from simulation data. The paper tests the claim on two colloidal systems: a single particle in an optical trap and a one-dimensional double-well phase-transforming chain. In both cases the learned internal variables collapse onto physically meaningful manifolds, and the predicted macroscopic trajectories match direct simulations on held-out loading protocols.","feed_headline":"Neural nets discover hidden variables for far-from-equilibrium materials","feed_subtitle":"New scheme learns internal variables and Markovian, thermodynamically consistent dynamics from particle trajectories.","key_machinery":"The carrying mechanism is the joint IB-VONNs training loop. An information bottleneck encoder of the form $\\alpha=\\rho(\\langle h(x)\\rangle)$ (with $h$ and $\\rho$ neural networks, and the ensemble average over realizations ensuring permutation invariance) extracts internal variables, while a decoder represents the conditional microscopic distribution. VONNs, built on Onsager's variational principle, represent the free energy density by an integrable neural network and the dual dissipation potential by a partially input convex integrable neural network, which strongly enforces thermodynamic consistency. The three loss terms, reconstruction log-likelihood and $L^2$ trajectory errors for $\\chi$ and $\\alpha$, are trained together so that the state variables are sufficient to predict the future and to characterize the microscopic distribution at every time.","core_discovery":"The central discovery is an architecture, IB-VONNs, that jointly learns an information-bottleneck encoder-decoder and a Variational Onsager Neural Network. The encoder maps an ensemble of microscopic configurations to internal variables $\\alpha$ that are permutation-invariant, macroscopic descriptors; the decoder reconstructs the non-equilibrium microscopic distribution $q(x(t)|\\chi(t),\\alpha(t))$ using Gaussian, Gaussian-mixture, or conditional normalizing flow models; and VONNs learn free energy and dual dissipation potentials so that the evolution equations for $(\\chi,\\alpha)$ are Markovian and thermodynamically consistent. The paper reports that, for overdamped Langevin dynamics, the method discovers internal variables that capture salient features of the microscopic distribution, that the reconstructed distributions agree with direct simulations even when multimodal, and that predicted macroscopic observables such as external force and mean strain have relative $L^2$ errors of roughly one to a few percent on test protocols, including unseen sinusoidal loading.","pith_inferences":["Beyond the paper, if the sufficiency assumption holds broadly, this method offers a general computational route from stochastic microdynamics to macroscale constitutive laws, potentially covering plasticity, damage, and active matter without modeler-chosen internal variables.","The paper chooses $X=Y$ in the information bottleneck, so the encoder is not explicitly trained to predict the future microstate; a variant that sets $Y$ to a future microstate could sharpen the Markovianity guarantee and is a natural next step.","The loss weights are hand-tuned and the paper notes this is crucial for success; automatic or Bayesian weighting would be needed before the framework becomes a black-box tool for general problems, which is an extension rather than a claim of the paper.","Because conditional normalizing flows are universal approximators, the same decoder architecture should transfer to non-colloidal and inertial systems, provided Onsager's variational principle is replaced by an appropriate variational formulation for those dynamics."],"forward_implications":["For inelastic materials, internal variables and their evolution equations could be learned directly from atomistic or particle simulations, replacing phenomenological choices such as the multiplicative kinematic decomposition of the deformation gradient.","The learned decoder provides a non-equilibrium analogue of the Boltzmann distribution, allowing one to sample microscopic states consistent with a given macroscopic state, which could support the inverse problem of material design.","Because the learned dynamics are constrained to be Markovian and thermodynamically consistent, the resulting continuum models are predictive beyond the training data, as demonstrated on unseen pulling protocols.","Using conditional normalizing flows as the decoder removes the need for prior knowledge of the microscopic distribution family, making the approach applicable to strongly multimodal, far-from-equilibrium distributions.","The framework naturally yields spatially non-local evolution equations through the Onsager structure, as shown in the phase-transforming chain, where the dynamics of one spring depend on its nearest neighbors."],"supporting_citations":[{"why":"Introduces the information bottleneck principle used to formulate the trade-off between predictability and complexity of the learned latent/internal variables.","marker":"[66]"},{"why":"Provides the deep variational information bottleneck approximation that makes the mutual information loss tractable with neural networks.","marker":"[1]"},{"why":"Supplies Variational Onsager Neural Networks, the method used to learn thermodynamically consistent and Markovian evolution equations.","marker":"[23]"},{"why":"Introduces deep sigmoidal flows, the conditional normalizing flow architecture used to represent arbitrary microscopic probability distributions.","marker":"[22]"},{"why":"Establishes the statistical mechanics framework connecting state variables to microscopic distributions and Markovian evolution, which the paper builds upon.","marker":"[28]"},{"why":"Shows that the permutation-invariant encoder architecture is a universal approximator of set functions, justifying the representation of internal variables.","marker":"[70, 75]"},{"why":"Provides the roadmap criteria for macroscopic models, which the paper refines into mathematical requirements for internal variables.","marker":"[68]"}],"fun_headline_variants":["Machine learning finds internal variables for non-equilibrium materials","AI discovers hidden variables and dynamics for non-equilibrium materials","Data-driven bridge: AI learns internal variables for non-equilibrium thermodynamics","IB-VONNs: learning internal variables with thermodynamic consistency"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The framework assumes that the current state variables, including the learned internal variables, completely determine the microscopic probability distribution at every instant, so that no memory of the past is needed.","fun_headline_variants_meta":{"raw":{"variants":["Machine learning finds internal variables for non-equilibrium materials","AI discovers hidden variables and dynamics for non-equilibrium materials","Data-driven bridge: AI learns internal variables for non-equilibrium thermodynamics","IB-VONNs: learning internal variables with thermodynamic consistency"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000827,"raw_usage":{"total_tokens":3657,"prompt_tokens":1027,"completion_tokens":2630,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":643,"completion_tokens_details":{"reasoning_tokens":2562}},"tokens_in":643,"tokens_out":2630,"duration_ms":18868,"temperature":1.0,"reasoning_tokens":2562,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T04:26:49.498891+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the residual mutual information between the future microscopic state and the past history of the state variables, conditioned on the current $(\\chi,\\alpha)$; if this residual fails to vanish for any finite-dimensional internal variable set, the central Markovian sufficiency premise is false.","supporting_citations":[{"cited_title":"Variational Onsager Neural Networks (VONNs): A thermodynamics-based variational learning strategy for non-equilibrium PDEs","cited_arxiv_id":null,"evidence_quote":"Supplies Variational Onsager Neural Networks, the method used to learn thermodynamically consistent and Markovian evolution equations."},{"cited_title":"Neural autoregressive flows","cited_arxiv_id":null,"evidence_quote":"Introduces deep sigmoidal flows, the conditional normalizing flow architecture used to represent arbitrary microscopic probability distributions."},{"cited_title":"A statistical mechanics framework for constructing nonequilibrium thermodynamic models","cited_arxiv_id":null,"evidence_quote":"Establishes the statistical mechanics framework connecting state variables to microscopic distributions and Markovian evolution, which the paper builds upon."},{"cited_title":"Roadmap on multiscale materials modeling","cited_arxiv_id":null,"evidence_quote":"Provides the roadmap criteria for macroscopic models, which the paper refines into mathematical requirements for internal variables."}],"review_version":1}