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Learning Physics from Data: a Thermodynamic Interpretation

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Pith's one-line read When learning the dynamics of a damped rigid body, adding dissipation removes the energy ambiguity that pure Hamiltonian trajectories leave behind.

desk verdict Worth reading for the Casimir-lifting observation, but the paper overgeneralizes from one special dissipation model and its numerical demo is an in-sample fit. read the letter →

arxiv 1909.01074 v3 pith:UH266D7A submitted 2019-09-03 physics.data-an cond-mat.stat-mechnlin.AO

Francisco Chinesta , Elias Cueto , Miroslav Grmela , Beatriz Moya , Michal Pavelka , Martin Sipka This is my paper · ORCID
classification physics.data-ancond-mat.stat-mechnlin.AO
keywords machinelearningthermodynamicsGENERICPoissonbracketCasimirinvariantsrigidbodydimensionalityreductiondissipation
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 argues that machine learning of physical dynamics should be understood as a form of thermodynamic reduction: extracting a pattern from data is a dissipative process driven by entropy, while reversible evolution is Poisson (Hamiltonian) propagation within a level of description. Using the GENERIC structure as its template, it shows that when learning a system with a non-canonical Poisson bracket from trajectory data, the energy can be recovered only up to the Casimir invariants of the bracket. If the observed dynamics also contains dissipation of the energetic Ehrenfest type, the Casimir ambiguity disappears and the full energy functional can be learned from the trajectory. The claim is demonstrated on a freely rotating damped rigid body, where kinetic energy is reconstructed from angular-momentum trajectories once $ au>0$. This matters because real systems typically dissipate, so the dissipative part of data can be used as a resource rather than a nuisance for identifying physics.

What carries the argument

The load-bearing object is the GENERIC evolution equation (General Equation for Non-Equilibrium Reversible-Irreversible Coupling), written as $\dot x_i = L_{ij}\,\partial E/\partial x_j + \partial\Xi/\partial x^*_i$ evaluated at $x^* = \partial S/\partial x$, which splits dynamics into a reversible Hamiltonian part generated by a Poisson bivector $L$ and an irreversible gradient part driven by a dissipation potential $\Xi$. For the rigid body the Poisson bracket is non-canonical, $\{F,G\} = -\boldsymbol m \cdot (\partial F/\partial \boldsymbol m \times \partial G/\partial \boldsymbol m)$, and the Casimir $|\boldsymbol m|^2$ is conserved by the reversible part. Adding the energetic Ehrenfest regularization, $\dot{\boldsymbol m} = \boldsymbol m \times \partial E/\partial \boldsymbol m - (\tau/2)\, L^T (\partial^2 E/\partial \boldsymbol m \partial \boldsymbol m)\, L\,(\partial E/\partial \boldsymbol m)$, keeps $|\boldsymbol m|^2$ fixed while dissipating kinetic energy, which makes the Casimir visible in the trajectory and therefore learnable.

What would settle it

Simulate a rigid body with the same quadratic energy but a different Casimir-preserving dissipation, for example Rayleigh damping proportional to angular velocity, fit the same quadratic Ansatz to the trajectory, and check whether the learned inverse inertia tensor converges to the exact one as the trajectory lengthens; if it does not, the identifiability result is specific to the energetic Ehrenfest regularisation.

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Extended reading notes

Core claim

The paper's central claim is that reversible dynamics alone carries no information about the Casimirs of its Poisson bracket, so any energy learned from a purely Hamiltonian trajectory is determined only up to a shift by a function of those Casimirs (for the rigid body, up to a multiple of $|\boldsymbol m|^2$). When the same system is observed with dissipation included, the Casimirs enter the dynamics and the full energy becomes identifiable. The authors verify this on the damped rigid body: with $ au=0$ the learned quadratic energy differs from the exact one by a Casimir shift, while for $ au>0$ the learned matrix of second derivatives converges to the exact inverse inertia tensor.

Load-bearing premise

The argument assumes the observed dissipation has the specific energetic Ehrenfest form of Eq. (23), that the quadratic Ansatz for the energy is correct, and that the Poisson bivector and timestep of the numerical scheme are known; if a real system dissipates through a different Casimir-preserving mechanism, the paper does not establish that the full energy remains identifiable.

Editorial extensions

If this is right

  • For any system with a non-canonical Poisson bracket, purely Hamiltonian trajectory data determine the energy only up to the Casimir invariants of the bracket; generative models that ignore this will return shifted energies.
  • Adding irreversible dynamics of the type considered in Eq. (23) makes the full energy identifiable, so dissipative data are not a nuisance but a resource for learning physics.
  • The thermodynamic viewpoint supplies the missing embedding between reduced and detailed manifolds: reduction is entropy-driven and the embedding can be constructed via the maximum-entropy principle, connecting manifold learning methods such as POD and LLE with GENERIC.
  • Learned reduced models with GENERIC structure can be integrated to predict future states on the reduced manifold and then lifted to the original data space, giving predictions in the rich space where validation is meaningful.

Reading between the lines

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

  • The identifiability claim is demonstrated for one parametric family of dissipation (the energetic Ehrenfest regularization); a natural extension would test whether other Casimir-preserving dissipation mechanisms, such as linear Rayleigh damping, also lift the degeneracy.
  • If dissipation lifts the energy ambiguity generally, then deliberately adding controlled dissipation to experimental protocols could become a practical identification technique for the energies of non-canonical systems.
  • The entropy-driven learning picture suggests that fluctuations in the database carry thermodynamic information about the learning dynamics itself; analysing those fluctuations could supplement deterministic trajectory fitting.
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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

2 major / 4 minor

Summary. This paper proposes a thermodynamic interpretation of machine learning, in which learning/reduction is viewed as a dissipative dynamics driven by entropy within the GENERIC framework, while reversible Hamiltonian evolution is interpreted as propagation within a level of description. The authors illustrate the idea with POD, where the Rayleigh quotient is interpreted as an entropy and eigendecomposition as a gradient flow, and with a rigid-body example in which the energy is learned from angular-momentum trajectories. The central new claim is that for dynamics with non-canonical Poisson brackets, the energy can be reconstructed only up to Casimirs of the bracket from reversible trajectories, but when dissipation is present the full energy becomes identifiable; this is demonstrated numerically for the energetic Ehrenfest regularization of Eq. (23).

Significance. If the general identifiability claim were established, the paper would provide a useful bridge between non-equilibrium thermodynamics and data-driven modeling: the Casimir obstruction is a real and often ignored issue in learning Hamiltonian systems, and the idea that dissipation of the specific Ehrenfest type reveals Casimirs is an interesting and testable observation. The paper ships reproducible code for the rigid-body experiment and states a falsifiable prediction (full energy identifiable with dissipation of the form (23)). The mathematical facts invoked about Casimirs and about the gradient-flow representation of eigenproblems are correct. However, the paper is primarily conceptual; it does not provide a general identifiability theorem, and the only quantitative evidence is a self-consistency check on data generated by the same model. The significance is therefore moderate: as a perspective piece with a concrete illustration, it is valuable; as a general result, it is incomplete.

major comments (2)
  1. [Sec. 6 and Sec. 5, Eq. (23)] The claim in the abstract, introduction, and conclusion that 'when having both the reversible and irreversible (dissipative) terms, the whole energy can be learned' is stated without qualification, but the demonstration only covers the energetic Ehrenfest dissipation of Eq. (23), where the dissipative term contains the Hessian of E and is therefore sensitive to Casimir shifts E -> E + λ |m|^2 because L^T L = |m|^2 I - m m^T. For a generic Casimir-conserving dissipation that is a gradient projected onto the sphere, e.g. dm/dt = m × grad E - γ (I - m_hat m_hat^T) grad E with γ > 0, the shift changes grad E by 2λ m, which lies in the kernel of the projector, so the observed trajectory is identical for all λ and the full energy remains unidentifiable. The paper itself states in Sec. 5 that the Poisson bivector, timestep, quadratic Ansatz, and dissipation mechanism are assumed known. The general conclusion is therefore not established; the authors should either prove an identifiability condition for the class of GENERIC dissipations under consideration or restrict the claim to the specific dissipative structure of Eq. (23).
  2. [Sec. 5] The numerical experiment is an in-sample self-consistency check: the trajectories are generated by integrating Eq. (23) with a known quadratic energy and known τ, and the learning algorithm fits the same parametric family to those same trajectories. The paper does not report any out-of-sample prediction, noise sensitivity, or cross-validation, so the demonstration supports identifiability of the parameters from perfectly observed data only, not the ability to learn energy from real noisy data. The text should state this explicitly (or add an out-of-sample experiment) so the reader does not infer a stronger claim than the experiment supports.
minor comments (4)
  1. [Sec. 5, paragraph after Fig. 2] The sentence 'By adding dissipation, the Casimirs now play a role in the dynamics, and the can be learned from the trajectory' contains a typo: 'the' should be 'they'.
  2. [Fig. 2 caption] The caption states both that the energy error 'eventually vanishes as the dissipation becomes significant' and that 'the higher the exact value of the coefficient (stronger dissipation), the higher is the error between the exact and learned values'; these statements appear contradictory, and the text should clarify whether the latter refers only to the error in τ and why that error increases.
  3. [Sec. 3.2.4] The entropy S(x) in Eq. (12) is introduced as an interpretation ('We shall interpret ...'), not as a derived quantity; stating explicitly that this is an analogy would help readers distinguish between the thermodynamic formalism and the POD algorithm.
  4. [Sec. 4] The free-surface-liquid illustration is a summary of previous work [12] with no new simulation; consider shortening it or clearly labeling it as a review so that the novel contribution of the paper is not obscured.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Sec. 5 fit is an identifiability test with known ground truth, not an out-of-sample prediction, and the Sec. 6 generalization is an extrapolation beyond the demonstrated dissipation family rather than a circular step.

full rationale

The paper's central numerical demonstration in Sec. 5 is an inverse-problem check: synthetic trajectories are generated from the quadratic energy Eq. (21)-(22) using the energetic Ehrenfest dissipation Eq. (23), and then the same parametric form is fitted by least squares. The paper explicitly states that the learned energy and tau are compared with 'the exact ones (used when generating trajectories).' This is not circular in the forbidden sense: the fit is not presented as an out-of-sample prediction, and the experiment is designed to show identifiability of the energy matrix when dissipation is present. The fact that the data are generated from the same ansatz is exactly what makes the ground-truth comparison meaningful; it does not force the claimed result, because with tau = 0 the same fit demonstrably fails to identify the Casimir direction. The mathematical mechanism is explicit in Eq. (23): the dissipative term contains L^T (d^2 E/dm dm) L dE/dm, so a Casimir shift E -> E + lambda m^2 changes the dissipative term even though it leaves L m = 0; the numerical fit then recovers the full Hessian. This is a valid demonstration for the stated model class. The paper's broader Sec. 6 wording ('the whole energy can be learned') goes beyond the one dissipation family demonstrated, since Sec. 5 assumes the Poisson bivector, timestep, quadratic Ansatz, and the Ehrenfest dissipation mechanism are known. That is a limitation of scope and a potential overgeneralization, i.e., a correctness or risk concern, not a circularity: no equation in the paper reduces to its own input by definition, and no fitted parameter is renamed as a prediction. The heavy reliance on the authors' own GENERIC and Ehrenfest literature ([14]-[17], [23], [28], [33], [37], [38]) is normal citation of previously published, externally falsifiable frameworks; in particular Eq. (23) is quoted from [38] rather than derived from the present data. Consequently no circular step meeting the required quote-and-reduction standard is present.

Assumptions & free parameters 4 free parameters · 6 assumptions · 2 invented entities

The central claim depends on the assumed GENERIC structure, the quadratic energy Ansatz, the specific dissipative regularization from [38], and the interpretation of the Rayleigh quotient as entropy. The only genuinely fitted quantities are the energy matrix entries and tau, but the demonstration is in-sample. No new physical entities are introduced; the invented entities are interpretive labels.

free parameters (4)
  • Energy matrix E_ij entries = not reported, only squared errors in Fig. 2
    In Sec. 5, the quadratic Ansatz E = 1/2 E_ij m_i m_j is fitted by curve_fit to the simulated angular-momentum trajectory; the values are not listed.
  • Dissipation coefficient tau = learned values in [0, 8e-3] (exact values used to generate data)
    Coefficient of the energetic Ehrenfest regularization in Eq. (23); estimated together with the energy matrix via least squares.
  • Number of retained POD modes k (and LLE neighbor count K) = not specified
    The reduced dimension k in POD and number of neighbors K in LLE are user-chosen; the paper does not give a selection criterion.
  • Time step of numerical scheme = not stated
    Explicitly assumed known in Sec. 5: 'The Poisson bivector and timestep of the numerical scheme are assumed to be known.'
assumptions (6)
  • domain assumption Reduced dynamics has the GENERIC form (3): dx/dt = L grad E + gradient of dissipation potential.
    The entire thermodynamic interpretation and the reduced vector field in Sec. 4.3 assume the GENERIC structure from the authors' prior work [14-17].
  • domain assumption Energy of the rigid body is a quadratic form in angular momentum (Eq. 21).
    True for rigid-body kinetic energy, but assumed a priori in the learning procedure rather than discovered.
  • standard math The Poisson bracket of the rigid body is the non-canonical bracket (19).
    Standard result of Poisson reduction, cited to [53-55]; the learning procedure assumes this bracket is known.
  • ad hoc to paper The Rayleigh quotient S(x) = x^T A x / x^T x is the entropy of the POD learning process.
    Introduced in Sec. 3.2.4 as an interpretation ('Let us interpret the Rayleigh quotient as entropy'); this identification is not derived from physical entropy.
  • domain assumption Dissipative term in Eq. (23) is the energetic Ehrenfest regularization with known structure.
    Borrowed from [38]; the general conclusion that dissipation lifts the Casimir degeneracy is shown only for this specific dissipation.
  • ad hoc to paper Data trajectories in Sec. 5 are generated by the same model that is subsequently fitted.
    The identification experiment is a self-consistency check on synthetic data, not a test on independent physical measurements.
invented entities (2)
  • POD entropy (Rayleigh quotient reinterpreted as thermodynamic entropy)
    purpose: to give a thermodynamic interpretation of dimensionality reduction by POD and LLE
    Sec. 3.2.4: S(x) = x^T A x / x^T x is declared to be the entropy driving the learning evolution; it is a mathematical analogy with no external falsifiable prediction.
  • Learning time evolution / learning dynamics
    purpose: conceptual bridge between GENERIC reduction and machine learning
    The paper reinterprets pattern recognition as a dissipative evolution driven by entropy; this is a viewpoint, not a measurable entity.

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Cite this review

Pith. "Pith review of Learning Physics from Data: a Thermodynamic Interpretation." pith.science (2026). https://pith.science/paper/UH266D7A

@misc{pith2026190901074,
  author       = {Pith},
  title        = {Pith review of: Learning Physics from Data: a Thermodynamic Interpretation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UH266D7A}},
  note         = {Machine review of arXiv:1909.01074}
}
read the original abstract

Experimental data bases are typically very large and high dimensional. To learn from them requires to recognize important features (a pattern), often present at scales different to that of the recorded data. Following the experience collected in statistical mechanics and thermodynamics, the process of recognizing the pattern (the learning process) can be seen as a dissipative time evolution driven by entropy from a detailed level of description to less detailed. This is the way thermodynamics enters machine learning. On the other hand, reversible (typically Hamiltonian) evolution is propagation within the levels of description, that is also to be recognized. This is how Poisson geometry enters machine learning. Learning to handle free surface liquids and damped rigid body rotation serves as an illustration.

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