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Bridging statistical mechanics and thermodynamics away from equilibrium: a data-driven approach for learning internal variables and their dynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2501.17993 v1 pith:3KLH6N2M submitted 2025-01-29 cond-mat.stat-mech math-phmath.MP

Weilun Qiu , Shenglin Huang , Celia Reina This is my paper · ORCID
classification cond-mat.stat-mechmath-phmath.MP MSC 82C3168T07
keywords non-equilibriumthermodynamicsinternalvariablesinformationbottleneckconditionalnormalizingflowsVariationalOnsagerNeuralNetworksoverdampedLangevindynamicscoarse-grainingmachinelearning
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 5 minor

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.

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 (3)
  1. [Section 2.2, Eq. (2) and Section 2.4, Eqs. (14)--(15)] 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.
  2. [Example 2, Eq. (51) and Figures 16, 20] 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.
  3. [Section 5 and abstract] 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.
minor comments (5)
  1. [Section 3.3, last paragraph] 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.'
  2. [Section 4.4, first paragraph and Figure 13 caption] 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'.
  3. [Section 2.2, paragraph after Eq. (4)] 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.
  4. [Section 4.3, Eq. (54)] 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.
  5. [General] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: held-out predictions are validated against external benchmarks; self-citations are used as building blocks, not as the evidence for the predictions.

full rationale

The paper's central claims are supported by independent empirical checks rather than by construction. The optical-trap example is tested against the analytic solution of the Fokker-Planck equation, and both examples are evaluated on held-out pulling protocols and initial conditions that were not used in training; the reported relative L2 errors are for these held-out cases. The VONN dynamics losses (Eqs. 14-15) fit a Markovian evolution law in the learned latent variables, and the Markovian form is an architectural assumption rather than a quantity derived from data, but the held-out protocol predictions provide genuine evidence that the fitted dynamics generalize. The information-bottleneck part is weakened by the paper's own admission that X and Y are chosen identical and the I(Z;X) term is dropped, so it functions mainly as a reconstruction autoencoder; the paper even notes in Section 2.4 that setting the reconstruction weight to zero would not affect the discovery of internal variables, which further shows the predictive content comes from the dynamics fit, not from a circular IB objective. There are load-bearing self-citations, especially to the authors' prior VONNs paper [23] for thermodynamic consistency and to the authors' prior STIV paper [28] for the theorem that exact reproduction of the microscopic distribution implies Markovian evolution; however, these are published prior constructions used as components, and the paper does not rely solely on them for its empirical conclusions. The main limitation, which is a correctness risk rather than a circularity, is that sufficiency of the learned internal variables is not fully tested at the joint-distribution level in Example 2, where only marginal distributions are validated (Figs. 16 and 20). Overall, no derivation step reduces to its own inputs by construction, so the paper is not significantly circular; the minor self-citation usage justifies only the lowest non-zero score.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The reconstruction-dynamics loop rests on the assumed existence of a finite-dimensional sufficient statistic and on several user-chosen training parameters. External benchmarks (analytic solution and held-out simulation data) keep the circularity burden low, but the theoretical guarantee for the general case is not established.

free parameters (4)
  • Loss weights lambda_pdf, lambda_epsilon/lambda_Fex, lambda_alpha = Example 1: 1e-5, 1, 1; Example 2 GMM: 0.002, 1, 1; Example 2 CNF: 0.001, 1, 1
    Hand-tuned; the authors state that no automatic weighting method works sufficiently and that the choice is crucial for training (Sections 2.4, 4.3).
  • Number of internal variables = One (Example 1), one per spring (Example 2)
    User-specified rather than learned; the paper states one per spring is 'sufficient to obtain a first order approximation' (Section 4.1), which limits the claim of automatic discovery.
  • Characteristic scales f* and phi* = Ex1: f* = sigma_Fex * sigma_lambda, phi* = f*; Ex2: f* = sigma_Fex * sigma_epsilon * L0, phi* = f*
    Estimated from data to normalize the neural potentials; they affect conditioning and the resulting loss landscape (Sections 3.3, 4.3).
  • Network architecture sizes and CNF hidden dimension d = INN/PICINN 2 layers x 20; encoder 2x10; decoder 2x10; CNF d=2
    Chosen by hand and part of the complexity control, since the I(Z;X) term is dropped from the variational IB objective (Sections 2.2, 3.3, 4.4).
assumptions (5)
  • domain assumption Overdamped Langevin dynamics describe the microscale for both examples
    The method is formulated for particle systems whose microscopic evolution is overdamped Langevin (Eqs. 17 and 36); all empirical validation is within this class.
  • domain assumption The instantaneous state variables (chi, alpha) fully determine the microscopic probability distribution
    Stated in Section 2.1 as a requirement; this Markovian embedding assumption justifies learning Markovian evolution, but its existence is not proven for Example 2.
  • domain assumption Free energy and dissipation potentials exist as local densities, with no gradient term in Example 2
    The paper assumes F = sum f_i and D = sum psi_i with f_i = f(epsilon_i, alpha_i) and psi_i = psi(epsilon_i, alpha_i, v_i, alpha_dot_i) (Section 4.2), explicitly omitting phase-field gradient terms.
  • ad hoc to paper The IB target Y is set equal to the same-time microstate X
    Section 2.2: 'X and Y are chosen to be identical'. This converts the information bottleneck into a same-time autoencoder objective and does not directly optimize predictive information about the future.
  • ad hoc to paper Per-spring marginal distributions suffice to characterize the microscopic distribution in Example 2
    The reconstruction loss (Eq. 51) sums log q(epsilon_i | z_i) over springs, so the joint distribution across springs is never modeled or tested.
invented entities (1)
  • Learned internal variable alpha independent evidence
    purpose: Augments the equilibrium state variables to remove memory effects and to characterize the non-equilibrium microscopic distribution
    Alpha is a latent construct computed as rho of the ensemble average of h(x) (Eq. 1). It has a falsifiable handle: test-set dynamics predictions and reconstruction of microscopic marginals are checked against independent simulation data. It is not claimed to be a new physical particle or field, but a learned coarse-grained descriptor.

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Cite this review

Pith. "Pith review of Bridging statistical mechanics and thermodynamics away from equilibrium: a data-driven approach for learning internal variables and their dynamics." pith.science (2026). https://pith.science/paper/3KLH6N2M

@misc{pith2026250117993,
  author       = {Pith},
  title        = {Pith review of: Bridging statistical mechanics and thermodynamics away from equilibrium: a data-driven approach for learning internal variables and their dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3KLH6N2M}},
  note         = {Machine review of arXiv:2501.17993}
}
read the original abstract

Thermodynamics with internal variables is a common approach in continuum mechanics to model inelastic (i.e., non-equilibrium) material behavior. While this approach is computationally and theoretically attractive, it currently lacks a well-established statistical mechanics foundation. As a result, internal variables are typically chosen phenomenologically and lack a direct link to the underlying physics which hinders the predictability of the theory. To address these challenges, we propose a machine learning approach that is consistent with the principles of statistical mechanics and thermodynamics. The proposed approach leverages the following techniques (i) the information bottleneck (IB) method to ensure that the learned internal variables are functions of the microstates and are capable of capturing the salient feature of the microscopic distribution; (ii) conditional normalizing flows to represent arbitrary probability distributions of the microscopic states as functions of the state variables; and (iii) Variational Onsager Neural Networks (VONNs) to guarantee thermodynamic consistency and Markovianity of the learned evolution equations. The resulting framework, called IB-VONNs, is tested on two problems of colloidal systems, governed at the microscale by overdamped Langevin dynamics. The first one is a prototypical model for a colloidal particle in an optical trap, which can be solved analytically, and thus ideal to verify the framework. The second problem is a one-dimensional phase-transforming system, whose macroscopic description still lacks a statistical mechanics foundation under general conditions. The results in both cases indicate that the proposed machine learning strategy can indeed bridge statistical mechanics and thermodynamics with internal variables away from equilibrium.

Figures

Figures reproduced from arXiv: 2501.17993 by the authors.

Figure 1
Figure 1. Overview of the IB-VONNs framework. It is composed of an encoder-decoder architecture that relates the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. A schematic plot of the encoder architecture, used to learn the internal variables [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. (a) A schematic plot of the CNFs decoder architecture (with [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: A schematic plot of the IB-VONNs architecture. It can be understood as an IB structure to learn the internal [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: A prototypical model for a colloidal particle in an optical trap. The particle is connected to a fix point via a [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: (a) Decoder predictions of the mean particle’s displacement versus the ground truth (directly computed from [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Comparison between the analytical values and the VONNs predictions of (a) the free energy density [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Comparison between the test data and the VONNs predictions of (a) the external force [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: (a) Data distribution in the space of state variables [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: Comparison between the data generated by a smoothed linear pulling protocol [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: The interparticle potential in Example 2. [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: A schematic plot of the IB-VONNs architecture. As shown in the middle part of the figure, the evolution of [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Results from the IB-VONNs method with CNFs. (a) The distribution of training data and the data generated [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: IB-VONNs with CNFs model tested on data with initial condition [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: Predictions of the standard deviation of the displacement by the decoder versus the ground truth for the [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]
Figure 16
Figure 16. Figure 16: Microscopic marginal distributions q(˜εi(t)|z(t)) computed by the reconstruction network with initial condition Ltot = 0.17 (test set) at t = 1.0. The histograms are computed from data, the solid lines represent the reconstructed probability distributions by IB-VONNs …
Figure 17
Figure 17. Figure 17: Prediction of the evolution of (a) the mean strain and (b) the internal variable with the IB-VONNs framework [PITH_FULL_IMAGE:figures/full_fig_p024_17.png]
Figure 18
Figure 18. Figure 18: Results from the IB-VONNs method with GMM. (a) The distribution of training data and the data generated [PITH_FULL_IMAGE:figures/full_fig_p025_18.png]
Figure 19
Figure 19. Figure 19: IB-VONNs with GMM tested on data with initial condition [PITH_FULL_IMAGE:figures/full_fig_p026_19.png]
Figure 20
Figure 20. Figure 20: Microscopic distributions q(˜εi(t)|z(t)) computed by IB-VONNs with CNFs with initial condition Ltot = 0.17 (test set) at t = 1.0. The histograms are computed from data, and the solid lines are the prediction of CNFs, and the dashed lines are the equilibrium marginal d…
Figure 21
Figure 21. Figure 21: Dynamics predictions of (a) the mean strain and (b) the internal variabl with the IB-VONNs framework with [PITH_FULL_IMAGE:figures/full_fig_p028_21.png]

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Forward citations

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    cond-mat.stat-mech 2025-06 conditional novelty 6.0 of 10

    A closure method that turns overdamped Langevin dynamics into gradient-flow macroscopic models for arbitrary approximate densities, automatically satisfying the second law.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.