Thermodynamic Interpolation: A generative approach to molecular thermodynamics and kinetics

Feliks N\"uske; Mathias Schreiner; Selma Moqvist; Simon Olsson; Weilong Chen

REVIEW 3 major objections 5 minor 15 references

Thermodynamic Interpolation: A generative approach to molecular thermodynamics and kinetics

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Thermodynamic Interpolation learns a differentiable map between Boltzmann ensembles at different temperatures, and the paper claims this map generates accurate ensembles, free energies, and approximate kinetic rates at temperatures not…

desk verdict A promising temperature-interpolating flow method, but the printed importance weight in Eq. (20) has both energy terms negative, which breaks the unbiased sampling construction as written. read the letter →

arxiv 2411.10075 v1 pith:6G2YL7W3 submitted 2024-11-15 physics.chem-ph

Selma Moqvist , Weilong Chen , Mathias Schreiner , Feliks Nüske , Simon Olsson This is my paper · ORCID

classification physics.chem-ph

keywords ThermodynamicInterpolationnormalizingflowsBoltzmanngeneratorsfreeenergyperturbationgEDMDkineticstemperaturegeneralizationstochasticinterpolants

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

Thermodynamic Interpolation (TI) is a generative method for mapping samples between Boltzmann distributions at different temperatures. The paper claims that a learned diffeomorphism, trained on molecular configurations at a few temperatures, transforms high-temperature ensembles into accurate low-temperature ensembles at temperatures never seen in training, with high effective sample size. It further claims that these transformed samples yield accurate free-energy differences via targeted free energy perturbation, and that combining TI with generator extended dynamic mode decomposition (gEDMD) gives approximate kinetic rates whose temperature dependence matches reference simulations. The goal is to make one learned model stand in for many costly low-temperature simulations while giving both thermodynamics and kinetics across a temperature range.

What carries the argument

The central object is the learned diffeomorphic map $f_{AB}$ between Boltzmann distributions $\mu_A$ and $\mu_B$ at temperatures $T_A$ and $T_B$, implemented as a continuous normalizing flow with an SE(3)-equivariant ChiroPaiNN velocity field $b^{(\theta)}(t,x;T_A,T_B)$ trained by the stochastic interpolant objective. The two-sided interpolant $x(t) = (1-t)x_A + t x_B + \gamma(t) z$ defines a stochastic path between paired samples, and minimizing the regression loss (eq. 19) yields a velocity whose integrated ODE transports $x_A \sim \mu_A$ to $x_B \sim \rho_B$. The same flow supplies the log-Jacobian determinant entering the importance weights $w_a$ (eq. 20), which correct the surrogate samples to unbiased estimates at the target temperature. For kinetics, the machinery is the gEDMD estimator of the Koopman generator for overdamped Langevin dynamics, whose eigenvalues give relaxation rates, applied to the TI-generated equilibrium samples.

What would settle it

Run long unbiased underdamped molecular dynamics for N-methylformamide at 300 K, build a Markov state model, and compare its slowest relaxation rate with the gEDMD rate obtained from TI-generated samples at the same temperature; a substantial mismatch would localize the failure in the Brownian-generator assumption rather than the learned map.

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Extended reading notes

Core claim

The central claim is that a continuous normalizing flow can learn a velocity field, trained by stochastic interpolants, that maps a Boltzmann ensemble at one temperature directly onto another, and that this map generalizes to target temperatures outside the training set. On the model systems studied (a one-dimensional asymmetric double well and two molecules from the QM9 set, N-methylformamide and 3-propan-2-ylhex-1-yne), the ambient TI map produces torsion-angle histograms and TICA projections that match reference low-temperature MD, and Kish effective sample sizes remain high enough that free-energy differences from as few as five transformed samples match what direct reweighting needs roughly twenty times more samples to achieve. The latent TI variant, a temperature-conditioned Boltzmann generator, samples multiple temperatures on demand from a shared latent Gaussian. The authors also show that gEDMD, applied to TI-generated equilibrium samples, yields rates for slow metastable exchanges whose temperature trends agree with rates estimated from MD, provided the underlying dynamics are taken to be overdamped Langevin (Brownian) dynamics.

Load-bearing premise

The kinetic rate estimates assume the molecular dynamics follows overdamped Langevin (Brownian) dynamics, so the generator used in gEDMD is an approximation for real, underdamped molecules; if that assumption fails, the predicted rates are not the true transition rates.

Editorial extensions

If this is right

Within the trained temperature range, one aTI model can produce ensembles at arbitrary target temperatures, replacing multiple dedicated low-temperature simulations.
Free-energy differences between temperatures can be estimated from very few transformed samples, which is valuable when sampling is expensive.
TI-generated equilibrium samples feed directly into gEDMD, giving temperature-dependent kinetic rates without time-correlated trajectories.
The latent TI variant shares statistical information across temperatures, so a single model can generate on-demand samples at several temperatures with improved accuracy compared with separately trained models.

Reading between the lines

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

The same learned-map architecture should extend to thermodynamic variables other than temperature, such as pressure or chemical potential, since the flow is conditioned on the state parameters; this is a natural but untested extrapolation of the paper's approach.
Because the paper finds gEDMD rates are robust to mapping bias even without reweighting, kinetic studies might skip importance weights entirely, accepting a small systematic error for large computational savings; the paper hints at this but does not recommend it.
The high sample efficiency at very small sample counts suggests TI could be applied to expensive potential-energy surfaces, such as quantum-mechanical calculations, where generating many MD frames is prohibitive.
Multi-temperature training may reduce the amount of simulation data needed per temperature, mirroring the information sharing seen in multi-ensemble Markov models; the paper observes this effect but does not quantify data requirements.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this paper deserves a serious referee, but not as it stands. The core idea—temperature-conditioned stochastic interpolants that map directly between Boltzmann ensembles in ambient space—is genuinely new and worth building on. The held-out temperature experiments are the right kind of evidence: the models are trained on a range of temperatures and then predict a temperature left out, comparing against independent MD simulations. The double-well results show high ESS and accurate free energies with far fewer samples than direct reweighting, and the molecular torsion distributions match reference histograms well. The authors also cite related work fairly and acknowledge the main assumptions in the kinetics part.

Now the soft spots. The load-bearing problem is Eq. (20), the ambient-TI importance weight. As printed, it is exp(-E(x_A)/kT_A - E(f_AB(x_A))/kT_B + log|det J|). Both energy terms are negative. A simple change-of-variables calculation gives the correct unnormalized target/source ratio as exp(-E(f(x))/kT_B + E(x)/kT_A + log|det J|)—the first energy term must be positive. The identity limit is decisive: set T_A = T_B and f = id; the correct weight is 1 (up to a normalization constant), while Eq. (20) gives exp(-2E/kT), which preferentially downweights high-energy configurations and is not the Boltzmann measure. Since this equation is the basis for the reported ESS values and for unbiased observables from aTI, the central claim is not reproducible from the manuscript as written.

Other issues are smaller but still worth raising. The abstract and conclusion claim extrapolation beyond the training-temperature range, but every held-out temperature in the experiments lies inside the training interval. That is interpolation, not extrapolation. The IQR filtering of free-energy estimates is a post-hoc step that can bias the estimator if not handled carefully; the paper should justify it more rigorously. And the kinetic rates depend on the overdamped Langevin generator, an approximation the authors do flag, but the approximation is not validated against underdamped reference rates in the main text.

Who is this for? Computational chemists and physicist working on generative models for molecular ensembles. The idea is plausible and the experimental setup is honest, so the sign error looks like a fixable typo rather than a fundamental flaw. I would send it to peer review with the expectation that the authors correct Eq. (20), release code, and temper the extrapolation claims. Desk rejection would be premature.

Referee Report

3 major / 5 minor

Summary. This manuscript introduces Thermodynamic Interpolation (TI), a generative method for sampling molecular configurations across thermodynamic states. Two variants are presented: ambient TI, which learns a direct diffeomorphic map between Boltzmann ensembles at different temperatures in configuration space, and latent TI, which maps from a normal latent distribution to a temperature-conditioned ensemble. The velocity fields are trained with stochastic interpolants and SE(3)-equivariant message passing networks. The authors validate TI on a 1D asymmetric double well and two small molecules (N-methylformamide and 3-propan-2-ylhex-1-yne), reporting high effective sample sizes, accurate free energy differences via targeted free energy perturbation, and temperature-dependent kinetic rates obtained from generator EDMD. The paper claims interpolation and extrapolation in temperature, including temperatures not seen during training.

Significance. If fully correct, the TI framework would be a useful addition to the Boltzmann generator toolbox: it shares information across temperatures, avoids replica ladders, and connects generative sampling to TFEP and gEDMD. The paper's strengths are its grounding in the stochastic interpolant framework, the use of standard TFEP relations, and held-out temperature validation against independent MD simulations and numerical integration. Experimental comparisons are plausible, and the authors are transparent about some limitations (Jacobian cost, Brownian generator assumption). However, the central importance-weight formula for ambient TI contains a sign error that invalidates the unbiasedness argument as written, so the quantitative claims must be re-evaluated. The kinetics section is also explicitly model-dependent and should be framed as Brownian dynamics rates rather than true molecular rates.

major comments (3)

[Methods, Ambient TI, Eq. (20)] The importance weight in Eq. (20) is printed with two negative energy terms. A change-of-variables derivation from xA ~ mu_A to y = f_AB(xA) gives a target/source density ratio proportional to exp(-E(y)/(kTB) + E(xA)/(kTA)) times |det J|, so the first energy term must be positive. In the identity limit TA=TB and f_AB=id, the correct weight is 1 up to the usual partition-function constant, whereas Eq. (20) yields exp(-2E(x)/(kT)), which after normalization preferentially weights low-energy conformations and is not the target Boltzmann measure. Since Eq. (20) is the basis for the reported aTI ESS values and for unbiased observable estimates from ambient TI, the central sampling claim is not supported by the manuscript as written. The formula must be corrected and the numerical results re-examined.
[Methods, gEDMD; Results, Generator Estimation] The kinetic-rate estimates are derived under the overdamped Langevin generator in Eqs. (24)-(25), and the text states "assuming the generator is Brownian." For the molecular systems, the reference MD is underdamped, so the estimated rates are not true molecular transition rates; the paper's own MSM comparison for the ADW system shows that overdamped rates bound underdamped rates from below. The claims about "kinetic exchange rates" for molecules should be explicitly restricted to the Brownian model, and the comparison to MD-derived rates obtained with the same misspecified estimator does not validate true kinetics. This is a caveated but load-bearing limitation for the kinetics section.
[Results, Figures 3 and 4; IQR filtering] The authors state that they use an IQR-filtering strategy to "avoid high variance on the free energy estimators" (Figures 3B-E and 4B-C). If outliers are removed based on the values of the estimator or the importance weights, this can bias the reported free energies and effective sample sizes. The Supporting Information must specify the filtering rule, clarify whether it depends on the estimator, and demonstrate (e.g., by reporting unfiltered or alternatively robust estimates) that the reported results are not an artifact of the filter.

minor comments (5)

[Abstract and Introduction] The term "Thermodynamic Interpolation" is introduced twice, and the sentence "We introduce TI... We introduce TI flavors" is redundant; please tighten the wording.
[Results, first paragraph] The molecule is referred to as both "3p2y1y" (abstract and Figure 5) and "3p2y1p" in the main text; please make the abbreviation consistent.
[Methods, Eq. (21)] The normalized temperature \hat T is defined with an overline T that is not explicitly defined; please define all symbols in the equation.
[Conclusion] The statement that exact computation of free energies "is not tractable in the current architectures" for larger systems is a strong limitation; please quantify the scaling for the tested systems or add a reference to the stated linear Jacobian cost.
[Manuscript formatting] The "TOC Graphic" appears twice on the final pages, which appears to be a duplication in the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the held-out-temperature predictions are genuine external benchmarks, and the cited methods are independent tools rather than self-referential premises.

full rationale

The paper's central claims are the learned ambient/latent TI maps can generate samples at temperatures held out from training, that free energy differences can be estimated from those maps, and that kinetics can be approximated from equilibrium samples. These claims are validated against independent references: for the asymmetric double well, free energies are compared to numerical integration of the Boltzmann distribution, and for the molecular systems, generated histograms and effective sample sizes are compared to reference MD simulations. The aTI models are explicitly trained with the target temperature excluded from the training set, so the reported results are genuine predictions rather than fitted quantities. The importance weights in Eq. (20) define the unbiased estimator, but the empirical ESS values and free energy estimates are benchmarked externally rather than being forced by construction. The paper does contain a separate internal reproducibility issue: Eq. (20) as printed has a sign error in the source-energy term, so at TA = TB with the identity map it yields weights proportional to exp(-2E/kT) instead of unity; however, this is a correctness/reproducibility defect, not a circularity in the derivation chain. Self-citations appear for the ChiroPaiNN architecture and for gEDMD/RFF estimators, but these are methodological tools from prior work by overlapping authors, not uniqueness theorems or ansatz assumptions used to force the present conclusions; they are externally established and machine-implementable. The overdamped Langevin generator assumption for kinetics is explicitly stated and caveated by the authors, and rates from underdamped simulations are also discussed. No step in the paper reduces, by definition or by fitted input, to the quantity it claims to predict. The derivation chain is therefore self-contained against external benchmarks, and the appropriate circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The method rests on standard statistical mechanics and flow-based generative modeling assumptions. No new physical entities are introduced. The main free choices are hyperparameters (temperature embedding length scale, noise schedule, gEDMD kernel parameters) and a post-hoc outlier filtering rule; the learned network weights themselves are fit to simulation data.

free parameters (4)

l0
Temperature embedding scale in eq (21); chosen by hand, controls how similar temperature embeddings are, affecting generalization.
gamma(t) noise schedule
Noise schedule in the two-sided interpolant (eq 18); specified in SI, affects the stochastic paths and learned velocity field.
gEDMD RFF bandwidth sigma and feature count p = sigma=0.6, p=50
Selected via VAMP score in Fig 6A; affects kinetic rate estimates.
IQR filter threshold
Outlier filtering for free energy estimators; a post-hoc data selection parameter mentioned in figure captions.

assumptions (5)

standard math The canonical Boltzmann distribution mu(x) = Z^{-1} exp(-E(x)/(kT)) is the target distribution for each thermodynamic state.
Used throughout; standard statistical mechanics.
domain assumption There exists a smooth invertible map (diffeomorphism) between Boltzmann distributions at different temperatures that the CNF can approximate.
Assumes stochastic interpolant training yields a valid velocity field with correct Jacobian log-determinants, as needed for TFEP weights.
domain assumption Molecular dynamics can be approximated as overdamped Langevin dynamics (eq 24) so that the gEDMD generator (eq 25) yields kinetic rates.
The kinetics results depend on this; the paper acknowledges real MD is underdamped.
domain assumption The learned models are trained on simulation data from a subset of temperatures and generalize to held-out temperatures.
Interpolation and extrapolation claims rest on this; no formal generalization guarantee is provided.
domain assumption SE(3)-equivariance is sufficient for molecular systems because chirality is conserved in classical MD data.
Justifies using ChiroPaiNN instead of an E(3)-equivariant architecture.

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Cite this review

Pith. "Pith review of Thermodynamic Interpolation: A generative approach to molecular thermodynamics and kinetics." pith.science (2026). https://pith.science/paper/6G2YL7W3

@misc{pith2026241110075,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic Interpolation: A generative approach to molecular thermodynamics and kinetics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6G2YL7W3}},
  note         = {Machine review of arXiv:2411.10075}
}

read the original abstract

Using normalizing flows and reweighting, Boltzmann Generators enable equilibrium sampling from a Boltzmann distribution, defined by an energy function and thermodynamic state. In this work, we introduce Thermodynamic Interpolation (TI), which allows for generating sampling statistics in a temperature-controllable way. We introduce TI flavors that work directly in the ambient configurational space, mapping between different thermodynamic states or through a latent, normally distributed reference state. Our ambient-space approach allows for the specification of arbitrary target temperatures, ensuring generalizability within the temperature range of the training set and demonstrating the potential for extrapolation beyond it. We validate the effectiveness of TI on model systems that exhibit metastability and non-trivial temperature dependencies. Finally, we demonstrate how to combine TI-based sampling to estimate free energy differences through various free energy perturbation methods and provide corresponding approximated kinetic rates estimated through generator extended dynamic mode decomposition (gEDMD).

Figures

Figures reproduced from arXiv: 2411.10075 by the authors.

**Figure 1.** Illustration of the two thermodynamic interpolation approaches. In the ambient case (green) one begins with Boltzmann distributed samples xA ∼ µA at the thermodynamic state A. These are transformed the into samples f (θ) AB(xA) ∼ ρB drawn from the surrogate distribution at state B, using the learned aTI map f (θ) AB. Finally, when estimating observables, samples are weighed using importance weights w a so that they… view at source ↗

**Figure 2.** Thermodynamic Interpolant neural architecture.A Latent and ambient TI for high-dimensional molecular-type systems. In the latent case we are only interested in sampling at a single temperature TA and do not input the second temperature TB to the cPaiNN model. In the ambient case, both temperatures TA and TB are included, along with atom numbers z. B Ambient TI for low-dimensional systems. In the lower-dimensional ca… view at source ↗

**Figure 3.** Results for Asymmetric double well potential.A The energy landscape of the 1D asymmetric double well model system. B Effective Sample Sizes (ESS) for the aTI model compared to direct reweighing. Here, the free energy was estimated using ∆F (TFEP) . C Estimated differences in Helmholtz free energy ∆F for the ambient TI (aTI) model and using direct re-weighing compared to true reference values. D–E Heatmaps of relativ… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: N-Me ensemble, effective sample sizes, and free energies.A A visualization of the N-Me molecule with labels for the atoms i, j, k, l which form the torsion angle ϕijkl. B The Effective Sample Size (ESS) plotted against the target temperature TB. C Estimated differences…

**Figure 5.** Figure 5: 3p2y1y ensemble and kinetics without reweighing. A A visualization of the 3p2y1y molecule. B The aTI torsion angles projected onto the first TICA (τ = 2 ps) dimension tIC1, plotted in a histogram. Along with the torsion angles, we also show reference MD values at the …

**Figure 6.** Figure 6: gEDMD analysis of Asymmtric Double Well system A VAMP scores22 as a function of the kernel bandwidth σ for different numbers of Fourier features p. We select p = 50 and σ = 0.6 as the model parameters. B Kinetic rates calculated under the Brownian assumption using gEDM…

**Figure 7.** Figure 7: Kinetic rate results estimated using gEDMD for N-Me, plotted against different target temperatures TB ∈ {300, 400, 500, 600, 700, 800, 900K} for (A) the latent TI (lTI), and the two ambient TI (aTI) approaches with initial conditions generated using lTI (B) and MD (C) …

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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