Boltzmann priors for Implicit Transfer Operators
Authors: Juan Viguera Diez
Mathias Jacob Schreiner
Ola Engkvist
Simon Olsson
Venue: ICLR 2025 Poster (2025)
Keywords: Molecular Dynamics
Generative Models
Transfer Operators
Diffusion Models
Boltzmann Generators
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We integrate Boltzmann Generators and Implicit Transfer Operator Learning to enable data-efficient training and interpolation between off-equilibrium and equilibrium models of Molecular Dynamics.

Abstract: Accurate prediction of thermodynamic properties is essential in drug discovery and materials science. Molecular dynamics (MD) simulations provide a principled approach to this task, yet they typically rely on prohibitively long sequential simulations. Implicit Transfer Operator (ITO) Learning offers a promising approach to address this limitation by enabling stable simulation with time steps orders of magnitude larger than MD. However, to train ITOs, we need extensive, unbiased MD data, limiting the scope of this framework. Here, we introduce Boltzmann Priors for ITO (BoPITO) to enhance ITO learning in two ways. First, BoPITO enables more efficient data generation, and second, it embeds inductive biases for long-term dynamical behavior, simultaneously improving sample efficiency by one order of magnitude and guaranteeing asymptotically unbiased equilibrium statistics. Furthermore, we showcase the use of BoPITO in a new tunable sampling protocol interpolating between ITOs trained on off-equilibrium simulations and an equilibrium model by incorporating unbiased correlation functions. Code is available at https://github.com/olsson-group/bopito.

Published as a conference paper at ICLR 2025
BOLTZMANN PRIORS FOR IMPLICIT TRANSFER OPERATORS
Juan Viguera Diez
Mathias Schreiner!
Ola Engkvist
Simon Olsson
Department of Computer Science and Engineering Chalmers University of Technology and University of Gothenburg SE-41296 Gothenburg. Sweden. 2 Molecular AI, Discovery Sciences, R&D, AstraZeneca Gothenburg, Pepparedsleden 1, 431 50 Mölndal, Sweden.
ABSTRACT
Accurate prediction of thermodynamic properties is essential in drug discovery and materials science. Molecular dynamics (MD) simulations provide principled approach to this task, yet they typically rely on prohibitively long sequential sim- ulations. Implicit Transfer Operator (ITO) Learning offers promising approach to address this limitation by enabling stable simulation with time steps orders of magnitude larger than MD. However, train ITOs, we need extensive, unbiased MD data, limiting the scope of this framework. Here, we introduce Boltzmann Priors for ITO (BoPITO) enhance ITO learning in two ways. First, BoPITO enables more efficient data generation, and second, it embeds inductive biases for long-term dynamical behavior, simultaneously improving sample efficiency by one order of magnitude and guaranteeing asymptotically unbiased equilibrium statistics. Furthermore, we showease the use of BoPITO in a new tunable sam- pling protocol interpolating between ITOs trained on off-equilibrium simulations and an equilibrium model by incorporating unbiased correlation functions. Code is available at htttps: /ggtttb..../...//ggg com/olssore group/bopitoo
INTRODUCTION
Efficient molecular dynamics (MD) simulation on long time-scales is critical to a large number of scientific and engineering applications. Stable simulations rely on solvers taking tiny integration time-steps, making the simulation of most phenomena impractical with current methods. Since these simulations are stochastic, a simulation step corresponds to drawing a sample from tran- sition probability density p(X(+r x:), where 7 is a tiny time-step. Recently, deep generative models have emerged as a promising strategy to potentially speed up these simulations by learning transition probability densities where T is much larger (Schreiner et al., 2023 Klein et al. 2023 Hsu et al. 2024; Fu et al.] (223)) and thereby allow efficient sampling at long time-scales. Implicit Transfer Operator (TTO) learning (Schreiner et al., 2023) learns such surrogate models at multiple time-resolutions. While ITO has shown promise in accelerating simulations, it relies on extensive unbiased simulation data which may not always be available. In Markov state models, this limita- tion can be mitigated by integrating off-equilibrium simulations and enhanced sampling simulations frendelkamp-Schroe et al. 2015 Rosta & Hummer 2014 Wu et al. 2014 2016). However, such estimators far unavailable for deep generative surrogates of the transition density.
Here, we introduce Boltzmann Priors for Implicit Transfer Operator (BoPITO) learning (Figure
BoPITO leverages pre- pre-trained Boltzmann Generators (BG) (Noé et al.) 2019: Viguera Diez et al. 2024 Klein & Noć, 2024 Köhler et al. 2023] Köhler et al. 2020 Midgley et al. 2024 2023) as priors to enable data-efficient training of ITO models, leading to one order of magnitude reduction in the simulation data needed for training. As the BG encodes the invariant measure or Boltzmann distribution of the dynamics encoded by the transition density, BoPITO, by construction, guarantees asymptotically unbiased equilibrium statistics. In this way, BoPITO can combine off-equilibrium data and biased data encoded into a BG prior train ITO models that predict MD across multiple time-scales. Using BoPITO we introduce a new sampling strategy to recover approximate dynamics from biased off-equilibrium data, BG prior, and unbiased time-correlation data, providing new method for inverse problems for molecular systems.
"maiil: simonols @ @chalmersssee

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Equilibrium data
dyn
eq
p(x)
Interpolation
e4
X N
+ AN dyn
$6
0qq
Eigenfunction
ddy
Molecular Dynamics
X
X N+
Figure 1: Boltzmann Priors for Implicit Transfer Operators (BoPITO) leverage pre-trained Boltzmann Generators to enable data-efficient training surrogate models of the transition den- sity after N simulation steps. BoPITO is implemented using Score-Based Diffusion Models. The score the model, ss, separates the contributions from the first eigenfunction of the transfer oper- ator (equilibrium density), Seq. from the rest, Sdyn- BoPITO embeds inductive biases for long-term dynamical behavior and enables interpolation between off-equilibruum and equilibrium models.
Our main contributions are
1. Boltzmann Priors for Implicit Transfer Operator Learning (BoPITO): We provide principled way to leverage Boltzmann Generators as priors for Implicit Transfer Opera- tor Learning, boosting sample efficiency, ensuring the generation of uncorrelated data and allowing asymptotically unbiased equilibrium statistics for oc.
2. BoPITO interpolators: tunable sampling protocol facilitating interpolation of BoPITO models trained on off-cquilibrium simulation data to an unbiased equilibrium distribution.
3. We show that BoPITO interpolators can recover approximate dynamics from models trained on biased simulations. We frame the optimization of interpolation parameters as an inverse problem and select the interpolated ensemble that is most consistent with unbiased observables.
BACKGROUND AND PRELIMINARIES
2.1 MOLECULAR DYNAMICS AND OBSERVABLES
Molecular dynamics (MD) is widely used simulation method in chemistry, physics, and biology. combines a mathematical model of the dynamics - e.g. Langevin dynamics (I (Langevin) 1908) with a potential energy model, U(x) Ω -> R. of system of interest, in turn providing detailed mechanistic insights of molecular systems, through the time evolution of particles in configuration space, E Ω. Practically, MD is solved by numerical integration, and time discretized, with step т. Consequently, we can understand MD as a Markov process with a Normal transition density p(x(+r xt), whose associated Markov operator has the Boltzmann distribution,
(xx) = - - (x))=) with 2 = =f ddx exp(-3U(x)),
(1)
as its invariant measure or stationary distribution, where 3 is the inverse temperature. One important application of MD compute expectations or observables Olsson 2022
1. Stationary observables:
O, E, [a(x)]-
2. Dynamic observables Time correlation functions:
Oa(t),b(++NT) = Ex,, Ex ((xx)))xxx+xxx+ b(x(+N+)]]
(3)
where Pr (X++N> |xx) is the conditional probability density after simulation steps with time-step
The maps a and b Ω + R serve as observable functions 'forward models' characterizing microscopic observation processes, e.g. indicating whether drug is bound or not, or an interatomic distance, leading observables such binding affinities and off-rates of drug target protein.
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Unfortunately, stable MD simulations rely using time discretizations, т, the order of 10 whereas properties such as protein folding or ligand unbinding occur on much longer time scales, 10 10-1s s. Due to the sequential nature of these simulations, simulation steps are necessary to predict these properties in an unbiased fashion. As a result, most MD data will be 'off-qquilirrimm e.g. trajectories exploring one a few of the modes of the Boltzmann distribution μ(x). See Appendix A.1 for more precise definitions of configuration space, off-equilibrium, unbiased and biased MD.
impractical number of
2.2 BOLTZMANN GENERATORS
Boltzmann Generators (BG) Noe al. 2019) are a generative machine learning approach to draw samples i.idd from the Boltzmann distribution, ,xx),, of physical many- systems (eq. [T.. BGs learn a diffeomorphic map, Fo RRN. R.N, from a latent space, equipped with a simple base density p(z), to configurational space of a physical system, such that the push-forward density p(x) = #p() closely approximates the Boltzmann distribution (eq. [. In practice, Fe is im- plemented using invertible neural network architecture with tractable Jacobian determinants to enable efficient sampling and exact sample likelihood computation (Chen et al. 2018 Papamakarios et al. 2021 BGs are trained either with approximately equilibrated simulation data with biased simulations, from i.e. enhanced sampling. by employing appropriate reweighting (Ferrenberg & Swendsen 1989 Shirts & Chodera 2008). Unbiased samples can then be generated by importance re-sampling (Nicoli et al.) (020). Similarly, unbiased expectations can be computed using the im- portance weights wxx) = e (x) /p()). As such, BGs are surrogates of the target Boltzmann distribution, but do not model time-correlation statistics.
2.3
TRANSFER OPERATORS
Transfer operators (Ruelle, 1978, Schütte et al. 2009) provide a framework describe the evolution of probability densities over time. Let p denote an initial probability density function on Ω, and its p-weighted version, p pp. The Markov operator, Tp. defined using a transition density p(X++r Xt).
[T(T) p)(x++)) p(x+++) / ((x dxe.
This operator describes the w-weighted evolution of absolutely convergent probability density func- tions on 52 by discrete-time increment 7 given by the dynamics encoded in a transition density p(x++r/xx ).). In the context of molecular dynamics, Ta is j-weighted equivalent to the Markov operator discussed above (sec. 2.Tb. The spectral form of the transfer operator is expressed as
(Ta(T) o p (x++T) = (=) (x(x++7);
(5)
where (T) are the eigenvalues, dy and 6, are the corresponding right and left eigenfunctions, respectively, and (flg) Sq (x))g(x) dx. The eigenvalues (x(T) depend on the parameter 7 and are related to the characteristic relaxation rates K; by [>x exp(-TR;). The right and left eigenfunctions are related by the stationary density, such that (x(x) p(x)oss In reversible
dynamics, all eigenvalues A; are real and lie within the interval -1 < X I Notably, there is one
eigenvalue X 1, with corresponding eigenfunctions th (x) and Φ1 (x) p(x).
2.4 DIFFUSION MODELS
Diffusion models (DMs) (Ho al. 2020 Song al. 2020) are popular generative modeling framework that approximates data densities p(x") by learning to invert a noising process ('forward diffusion').. The forward diffusion process pre-specified and incrementally transforms the data distribution into simple prior distribution, p(x"), through simulation of a time-inhomogenous Markov process represented by the SDE,
dx f(x faa lair) dlait + g(aai)) dW,
where (0 < tail < T) is the diffusion time, f and are chosen functions, and dW is a Wiener process. To generate samples from the data distribution, p(x²), sample from p(x") and
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solve the backward diffusion process ('denoising process") ") (Andrson 1982).
dx faift
= [ faits tdiit) - -g" (t V *tain 1o laiff (tait) dr + g(tai)) dW.
(7)
This backward process approximated using a deep neural network V Vxtau log pe ( (x"aat Iaar
(xfait, Adit). The learned backward SDE can be recast into probability flow ODE (Maoutsa et al. 2020; Song et al. (021), which in turn can be interpreted as a continuous normalizing flow (Chen 2018 which facilitates efficient sampling and enables sample likelihood evaluation.
2.5 IMPLICIT TRANSFER OPERATOR LEARNING
Implicit Transfer Operator (TTO) Learning (Schreiner et al. 2023) is framework for learning sur- rogate models of the transition density, p~ ( from MD data, where N is an arbitrarily large integer. ITO leverages that the transfer operator framework allows to express the transition den- sity as
p(X++NT = Σ (x)))
(8)
where th and d, are independent of T and N. See Appendix A.2 for complete derivation.
This decomposition inspired strategy to learn conditional generative model X Nr Po px( Xt, N) by sampling tuples (Xt;, N;r, N₂) and training a generative model to mimic the empirical transition density at multiple time-horizons, Nr. Here, we approximate ITO models with a conditional denoising diffusion probabilistic model (cDDPM) of the form
p p(x(+N,|x,,, N) =
p(x0 O.T p(x |x,, N)) dxdx"" dx!:T"
where X1.7 are latent variables of the same dimension as our output, and follow a joint density describing the backward diffuuson process, eq. 7 and x" (0, We denote diffusion time in Diffusion Models using superscripts, while physical time represented using subscripts. The conditional sample likelihood is given by
(T: 0) = II Po (x Xt, Nz) EII
(10)
where I list generated indices specifying a time 6; and a time-lag (r) integer multiple N,, associating two time-points in MD trajectory of length Mr, (Xo, Xy...., X(M-1)+)
Training is performed by optimizing an approximation of the variational bound the log- likellood Ho et al. 2020).
Setairr ( Ei (i.DI.rr [llc Eo :.+N. Xt,, N1, tai l|2]
(11)
33) and By is the variance of the forward diffusion process at diffusion time, j. éo(-) one of the architectures presented by Schreiner et al. (2023). Following Arts et al. (2023) we express the score as
where x
Vallait X
VI afaifee with afait II;
éo tatare + N, Xt,, Ni, laitt)
se N Xt, No, tditt)
(12)
VI afaii
See Appendix A.3 for more details and pseudo-code on training and sampling algorithms.
3 RELATED WORK
Sampling the Boltzmann distribution Apart from Boltzmann Generator-based approaches, there are number of traditional ways to draw statistical samples from the Boltzmann distribution of molecular systems. Prominent examples include, molecular dynamics or Markov Chain Monte Carlo simulations coupled with enhanced sampling strategies (Hémin et al.
2022 Kamenik al.
2022). including replica-base based approaches (Earl & Deem 2005| Sidler et al. 2016: Pasarkar et al,
2023
conformational flooding (Grubmüiler 1995). meta-dynamics (Laio & Parrinello 2002), and
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umbrella sampling (Torrie & Valleau 1977). in particular when paired up with machine-learnen col- lective variables (Chen & Ferguson 2018] Wang et al. (2019]]] Herringer et al. 2023; M. Sultan & Pande 2017). The success of these approaches relies on choosing the right mechanism to enhance sampling across high free energy barriers, or low probability regions, for any given case (Carter et al. [989). Finding such mechanisms typically involves substantial manual engineering of col- lective variables and other spper--arameters. Finally, there are transition path sampling approaches (Dellago et al. [998 Bolhuis et al. 22002), which are particularly powerful when combined with reinforcement learning (Jung et al. 2023) or deep generative priors (Plainer et al. 2023).
Latent space simulators and Coarse graining Classical (fine- grtee-gaaded) molecular dynamics sim- ulation can be coarse-grained beyond the Nomn-Oppenhemmer approximation, such that atomic nuclei are merged together into 'beads.' (Noid 2023) This strategy, in principle enables faster simulations due to the smaller number of particles and the acceleration of kinetics caused by the coarse- coarse-graining graining (CG) operation (Nüske et al., 2019). CG models can be estimated to closely approximate the ther- modynamies of the corresponding fine-grained system through the optimization of two equivalent variational bounds (Noid et al.) [0088] Lyubartsev & Laaksonen 1995; Ercolessi & Adams 1994). A relaxation of this bound was recently proposed to also allow for probabilistic reconstruction of fine- grained configurations (Chennakesavalu et al. 2023). These bounds have been used extensively to
build CG force-field models (Husic et al. 2020; Wang et al. 2019a Majewski et al.] 2023 Charron
et al. 2023) including implicit solvation models (Chen al. 2021 Katzberger & Riniker 2024),
using deep neural networks due to their ability to capture multibody terms (Wang et al.) (022)) to
ultimately accelerate the prediction of equilibrium properties of molecular systems. Similarly, the development of 'latent space simulators' where learned, typically low-dimensional, latent space equipped either with propagator (Sidky et al., 2020)] Chennakesavalu
al. 2023] Wang et al. 2024). or not (Wang& & Gómez-Bone Bombarelli 2211), is learned to enable efficient simulation. These approaches, in general aim to accelerate molecular simulations akin to BoPITO, yet due to the CG operation, the molecular dynamics (kinetics) will be accelerated, and detailed knowledge of the unbiased dynamics are needed to correct this (Nüske et al. 2019 Crommelin Vanden -ijndenn 2011]. Concurrent work, presents MDGen where conformational states are tokenized and in turn used to generate multiple frames of MD trajectory jointly (Jing et al., 2024).
Transfer Operator surrogates Analysis of MD data often involves building transfer operator
surrogates such as Markov state models (MSM) (Schütte et al., 1998 Prinz et al. 2011: Swope
al. 2004 Husic & Pande 2018), time-lagged independent components analysis (Molgedey &
Schuster] [ & Müller 1998 Pérez--ernändez et al.] 2013), Markov field models or dy-
namic graphical models, (Olsson & Noe Mardt et al. 2022 Hempel ct al. 2022). VAMPnets (Mardt al. 2018 Wu & Noë] 2019), or observable operator models (Wu et al. 2015). Markov state models are time-space discrete approximations of the transfer operator and Deep Generative MSM (Wu et al. 2019) and VAMPnets (Mardt et al., 2018) learn the space diseretization through deep neural networks. Dynamic graphical models or Markov field models (Hempel et al. 2022) represent time-space discrete approximation of the transfer operator injecting a (conditional) inde- pendence assumption of molecular subsystems, when modeling the transition probability, leading to better scaling for systems with poor time-scale separation OOssson & Noë 2019). Apart from ITO (Schreiner et al.) 2023), several other deep generative approaches for modeling the transition den- sity of molecular dynamics have recently proposed. Timewarp where a normalizing flow is used to encode the transition density (Klein et al., (023) with limited transferability to enable metropolized sampling of unbiased equilibrium distributions (Hastings) 1970). Score dynamics use a DDPM to model the displacements of initial configuration towards time-lagged one, achieving picosecond time-steps simulation and limited transferability (Hsu et al. 2024). However, unlike in the context of MSMs (Trendelkamp-Scheree are no deep generative transition density surrogates available leveraging available information about the equilibrium distribution BoPrTO is one such method.
al. 2015 Rosta & Hummer 2014 Wu et al. 2014 2016), there
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4 BOLTZMANN PRIORS FOR IMPLICIT TRANSFER OPERATOR LEARNING
ITO training requires extensive, unbiased MD simulations to capture the statistical distribution of rare events. This data-intensive requirement can hinder ITO's practical implementation. Further- more, models trained on off-equilibrium simulations, which may exhibit non-representative statis- tics, can lead to inaccurate predictions, compromising their utility in downstream applications. In practice, however, both unbiased off-equilibrium simulations and information about the equilibrium distribution from potentially biased simulations (Hénin et al., (022) can be cheaply generated, and both encode information about the dynamical behavior of molecular systems. We introduce Boltz- mann Priors for Implicit Transfer Operator (BoPITO), as a learning paradigm for ITO models that leverage available information about the equilibrium distribution.
BoPITO uses pre-trained models of the equilibrium distribution to improve ITO learning in four ways. First, it helps ensure broad sampling of proportional to the equilibrium distribution for subsequent MD simulations yielding information about the transition density p(x, | Xx) from across Q. Second, we use fix the stationary part of the learned transition density, boosting the sample efficiency when learning models of molecular dynamics. Third, it imposes inductive bias of long-time dynamical behavior allowing for recovery of unbiased Boltzmann distribution for long- time horizons. Fourth, using BoPITO, we introduce novel tunable sampling protocol interpolating ITO models trained on off-equilibrium simulation data and an unbiased equilibrium distribution.
4.1 EFFICIENT DATA GENERATION
Training data for ITO models consists of several independent unbiased MD simulations. Following the adaptive sampling strategy (Bowman et al. 2010 Doerr & De Fabritis) 2014) Viguera Diez et al.
2024 Betz& & Dror 2019), used extensively in the molecular dynamics simulation community, use a tre-tedined BG to generate initial conditions to simulations ensuring broad sampling across Ω proportional the Boltzmann distribution. As long as these trajectories reach 'local equilibrium' we can in principle recover an unbiased model of the molecular dynamics (Nüske et al. (017), albeit without relying on running one or few very long simulations reach the global equilibrium.
4.2 LONG-TERM DYNAMICS INDUCTIVE BIAS FOR ITO
One important application of ITO is to allow for one-step sampling of long-time-scale dynamics. However, real datasets often contain limited number of effective samples for long time-scales, leading to potential biases in models. To mitigate this issue, propose separating the equilibrium contribution from the time-dependent components:
p(x++N+|x;) = (ack)))=((x(++ ((x(+N+)) + X (T))
(13)
where we have used that A = 1 and its corresponding eigenfunctions are Φ1 1 and e 1. Additionally, we introduce a decay in the time-dependent component, creating an inductive bias that asymptotically samples from an available equilibrium model for long-term dynamics. Using the spectral decomposition of the transition density (eq.
13 we choose the score model as
6(x Xt, N, Lai, 0) = Sog (x (airr
A 8dyn (xfair N++ Xt, N, lairr O):
(14)
where Seq (onxxli dirr lair)) the score of pre-trained surrogate of the equilibrium distribution model, Å < 1 is hyper-parameter and Sayn 6xfa ( Xt, NN, tain, 0) accounts for the time-dependent components. As N + oc, Seq dominates and the model samples from the equilibrium model, Appendix A.4 for an example. By interpreting the score field as a velocity field, we can reformu- late the DM continuous normalizing flow which we can use for Metropolized sampling of the unbiased equilibrium distribution (Klein et al. 2023). The proposed factorization is principled, cor- responding to a separation of the score of the transition density into stationary and dynamic part, where the first part is considered known, see Appendix A.5 for details. In practice, we first train seq (if not provided) using equilibrium data. Then we train Sdyn with unbiased, possibly off-equilibrium, MD data while keeping Soq fixed and we choose performing hyper-paramett optimization, see A Appendix for details. We discuss alternative formulations of the score in Appendix A.7

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4.3 BoPITO INTERPOLATORS
A significant issue in MD simulations is the unbiased sampling of transitions over high free-energy barriers, e.g. channels in j((x) with very little probability mass, rypically coinciding with mixing on .2. This difficulty leads to an inability to predict time-correlation statistics for large Nr. With BoPITO we can define an interpolation between models trained on off-equilibrium simulations and the equilibrium distribution. In this manner, we can approximate dynamics not seen explicitly in the data. For time-dependent model, Sdyn: trained on off-equilibrium simulations with maximum training lag, N Nmxx. we define a BoPITO interpolator as model with the following score function:
s(xfue Xt, Ni,, tair) (x (tt fair)) + Nia sdyn (x X1, N fair);
(15)
with No 2 Max- This defines an interpolator because Nit = Nmax generates samples from the model distribution with maximum lag and Nint + oo generates samples from the equilibrium model.
Inspired by methods in molecular biophysics (Kolloff & Olsson 2023 Olsson et al. 2017| Bottaro & Lindorfi-Larsen 2018; Salvi al. (016), we propose choose the interpolation parameter, Nint as the most consistent with an unbiased dynamical observable, such as experimental data. That is, for a lag N > Nmax and an unbiased dynamic observable ON, we choose
Nin, = arg min JO ON..
(16)
Nia
where ON. is the dynamic observable estimated with samples from the BoPITO interpolator. This way BoPITO can integrate off-equilibrium and equilibrium MD with experimental data. In practice, find the interpolator to generate some high-energy states. However, energy structures by alternating long-lag interpolation steps with short-lag non-interpolation steps for local relaxation. We discuss how BoPITO interpolators alleviate -uittt-distiibutiin issues in Appendix A.8
alleviate the high-
RESULTS
For detailed parameters of the experiments below, refer to Appendix A.9
5.1 SYSTEMS
Prinz potential is ID potential commonly used for benchmarking MD sampling methods (Prinz al. 2011). We set the observable functions, a and b in eq. to be the identity function for computing dynamic observables. For details, see Appendix A.10
Alanine Dipeptide is small peptide with 22 atoms. We use publicly available data from Dibak et al. (2022, containing 1 jus simulation time split in 20 trajectories. Simulation is performed in an implicit solvent with Is integration time-step, and data is saved every ps. We choose,
sin d(x (x)
(x) b(x)
(17)
cos ((x)
where d(x) is torsion angle involved in the slowest transition in the system. For details, Appendix A.IT
Chignolin (cIn025) is fast folding protein with 10 residues, 166 atoms and 93 heavy atoms. We use molecular dynamics data previously reported by Lindorff-Larsen et al. (001T). The data is proprietary but available upon request for research purposes. The simulations were performed in explicit solvent with 2.5 fs time-step and the positions was saved at 200 ps intervals. We extract all heavy atoms positions from the simulations and train models on this data. We use the fraction of native contacts Madorff-Larseen et al. (2011) to define a dynamic observable with,
Σ
Nis EE
N; f>i
1004,y 4,
a(x) = b(x)
(18)
Σ N
where the first sum in the numerator iterates over all Nies residues in the protein, while the second sum considers the N; native contacts of residue separated by at least seven residues in the primary sequence. Here, (x) and di, represent the C. - C. distances of residues i and in the structure x and the native structure, respectively. This observable quantifies the protein's 's foldedness, with values ranging from 0 (unfolded) to 1 (folded). For detailed explanation, refer to Appendix A.12
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5.2 BOLTZMANN PRIORS FOR TRAINING DATA GENERATION
Modeling the transition density over Q requires observ- ing transitions across the state space. In the usual set- ting, only one or a few initial conditions are available
0.150
BG crystal
when data collection starts. Here, explore the case where a BG is available before data collection and com- pare it against the baseline, where only one initial condi- tion is known. We use a BG trained on equilibrium data of the Prinz potential sample the initial conditions of our training trajectories. We also generate trajectories using a single starting point (crystal, 0.75) and compare their performance with long simulations (Figure 2) We com- pare the two different data generation strategies using the metric, Acorrelation], which measures the absolute dif- ference of the coreecorelatioo function (dynamic observ- able) compared to the MD ground truth (Appendix 13).
correlation
0.125
0.100
0.075
0.050
0.025
10
20
30
40
60
70
Lag (steps)
(a)
We find that the performance of models using Boltzmann priors for data generation is superior for different lag times (Figure 21) and number of trajectories (Figure [ The gap between BG and the crystal baseline increases with lag-time, and as expected decreases with the number of generated trajectories.
BC crystal
0.20
correlation!
0.15
0.10
0.05
5.3 BoPITO EFFICIENTLY SAMPLES LONG-TERM DYNAMICS IN A LOW-DATA CONTEXT
50
100
150
200
Number of trajectories
Fixing the equilibrium contribution to the score field ef- fectively reduces the number of parameters that need to be estimated. To test whether this prior information man- ifests as an improved sample efficiency we compared ITO and BoPITO models against each other with varying sizes of training data.
(b)
Figure 2: Absolute difference in corre- lation with respect to long unbiased MD simulations (lower better) of mod- els trained on trajectories initialized on samples from a Boltzmann Generator (BG) and single structure (crystal, x 0.75) for the Prinz Potential under direct sampling. The former presents supe- rior performance for different lag times (number of trajectories 50) (a) and number of trajectories (b). The shaded areas correspond to 95 % confidence in- terval.
We find that the BoPITO models achieve higher ac- curacy for long-term dynamics compared to ITO mod- els when data is scarce, as the equilibrium distribution is known priori and does not need to be learned from sim- ulation data (Figure 3) The inductive bias in BoPPTO models enables them to learn long-term dynamics, even in scenarios where ITO models fail. For the Prinz Po- tential, we find that while ITO suffers from poor perfor- mance modeling long-term dynamics when data is scarce, BoPITO models accurately capture long-term dynamics without worsening the performance short and medium time-scales. The results for Alanine Dipeptide and Chignolin show favorable scaling with increasing system size: find that an order of magnitude more data is needed train ITO to the same accuracy as a BoPITO model trained on the same data. Moreover, we show in Appendix A.14 that the energies of configurations sampled from our prior Boltzmann Generator used in our experiments match closely the energies of MD samples.
5.4 INTERPOLATING BETWEEN MODELS TRAINED ON OFT-EQUILIBRIUM DATA AND THE BOLTZMANN DISTRIBUTION WITH EXPERIMENTAL DATA
In practical settings, our MD data will be off-equilibrium, e.g. having sampled only one or few of the relevant modes in the Boltzmann distribution μu() in a given trajectory. Consequently, unless extensive data across the domain S2 can be collected, models based on such data will be biased. An alternative to collecting more simulation data to use a multi-modal strategy where experimental data is used to fill the gaps left by simulation data and bridge to long time-scale dynamics (Salvi al., [016]] Kolloff & Olsson 2023). In this section we explore the potential of BoPITO inter- polators to integrate dynamic observables with off- efflequilirrium simulation data to recover unbiased long-term dynamics.
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Prinz potential
Alanine Dipeptide
Chignelin
0.12
short medium
|4 correlatin)
0.10
medium long
carrelation
0.4
medium
A Correla ation]]
0.4
0.08
0.3
ITO
0.06
0.2
0.2
0.04
0.92
0.1
0.1
0.00
16
0.0
10
20
30
40
10 Number of trajectories
15
Number of trajectories
Number of trajectories
0.12
short
correlation)
0:10
medium long
carrelation
0.4
medium
correlation attion]
0.4
medium
8.08
0.3
BoPITO
0.2
0.2
0.04
0.02
0.1
0.0
10
20
30
40
10 Number of trajectories
15
Number of trajectories
Number of trajectories
Figure 3: Absolute difference in correlation with respect to long unbiased MD simulations (lower is better, one-step sampling) of ITO (left) and BoPITO (right) split into short, medium, and long time- scales against the number training trajectories for the Prinz potential (top) and Alanine Dipeptide (middle) and Chignolin (bottom). The shaded areas correspond to 95 % confidence interval.
We showcase BoPITO interpolators on Alanine Dipep- tide. By removing the transitions between the modes of the Boltzmann distribution corresponding to the slowest process in the system, e.g. & crossing 0 or 2, generate a biased simulation data, resembling a realistic scenario. We then train ITO and BoPITO models using these biased trajectories. For lags >> 100, we sample the BoPITO in- terpolator estimated by matching to unbiased dynamic observable defined by eq. [26] allowing us to overcome a systematic error in the correlation function observed in the biased MD data and for an ITO model trained on these data (Figure
0.4
Correlation
0.2
MD
0.1
Blased MD
1060 1900
2200
ITO
0.0
BoPITO
10
10a
10a
10a
Lag time (ps)
Figure 4: Fitting an interpolator a bi- ased BoPITO model for Alanine Dipep- tide. Both the ITO and BoPITO models are trained on biased MD simulations of Alanine Dipeptide. We fit a BoPITO in- terpolator selecting the most consistent ensemble with the unbiased correlation function specified in eq. [6 Numbers with blue arrows specify the interpola- tion parameter, N.m.
Beyond reproducing the provided dynamic observable, the interpolator also demonstrates remarkable ability to capture the underlying microscopic dynamics (Figure 5). Even with excellent agreement, the generated ensembles only slightly overestimates density in the transition state region. However, this effect could easily be alleviated by annealing short MD simulation the interpolation was recently shown (Viguera Diez et al. 2024).
LIMITATIONS AND FUTURE WORK
Choice of scale of the dynamics after which the model is guaranteed sample equilibrium. In Appendix AA. we describe a protocol to determine a bound for this parameter. However, developing a similar protocol for transferable BoPTTO model would likely require modifications to accommodate the diversity of global relaxation time-scales across different systems.
Ā The hyper-parameter X, defines a global relaxation or mixing time-
No Shlapman-Kolmogonn Guarantee When training biased or off-equilibrium data where we rely on establishing ergodicity through interpolation we cannot guarantee self-consistency of the dynamics in the Koloonooooo sense.

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Surrogate model BoPITO inherits the current limitations of ITO, such generalization chemical space and thermodynamic variables, and scaling. Furthermore, current models cannot guarantee unbiased sampling dynamics for non-equilibrimm ensembles, which would require closed- form expressions for the target path probabilities.
107
0.5 ns
20"
30"
10
10:
10h
10'
1 ns
p(Xae|Xo)
[Xxo
20.1
10a
p(xaz Xa)
10
x(xax|Xo)
10-1
5 ns
20-1
20-1
7.5 ns
[XxarXXol (g)
(X
20
10-3
2013
Figure 5: BoPITO incorporate unbiased dynamic observables correct model trained biased data. Rows of increasing time-lag (from top to bottom). Contour plots correspond to a BoPTTO interpolator. The first column shows conditional transition densities projected onto the torsion angles á and sb (inset). The black cross indicates the initial condition. The second and third columns show marginal distributions of ó and respectively. MSM stands for Markov State Model of the unbiased MD data.
CONCLUSION
We introduce Boltzmann Priors for Implicit transfer Operator Learning (BoPITO), framework to enhance ITO learning in three ways. First, a broad sampling of configuration space is used to ini- tialize short off--quulibrium MD simulations. Second, we parameterize the transition density as an interpolation towards pre-trained Boltzmann Generator, improving sample efficiency by an order of magnitude, BoPITO is principled approach to embedding prior knowledge the stationary dis- tribution of Markovian dynamics as an inductive bias for long-term dynamical behavior. Third, our approach enables interpolation between models trained on off-equilibrium data and the equilibrium distribution, and observables. Consequently, BoPITO the first method to allow for the integration of multiple sources of information into the generation of deep generative surrogates of molecular dynamics.
recover accurate models of unseen dynamics when informed by unbiased
10

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ACKNOWLEDGMENTS
The authors thank Prof. Rocío Mercado and Prof. Fredrik Johansson for their useful feedback on the manuscript. This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation. The computations in this work were enabled by the Berzelius resource provided by the Knut and Alice Wallenberg Foundation at the National Supercomputer Centre.

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A
APPENDIX
A.1 DEFINITIONS
Configuration space: Mathematical space in which all possible states or positions of physical system are represented. For example, in a classical MD simulation, the positions of all the atoms in the simulation.
Unbiased simulations: Simulations performed with standard MD, i.c. Langevin Dynam- ics. They are unbiased because they generate samples from the underlying transition den- sity. Particularly, for asympiotically large simulation times, unbiased simulations sample the Boltzmann distribution p(x). However, because of energy barriers in the potential energy landscape, unbiased simulations may not explore some modes of the Boltzmann distribution if the simulation time not long enough. Therefore, generating samples with unbiased MD often leads to off-equilibrium data.
Off-equilibruum data: Simulation trajectories whose underlying statistics do not represent well those of the equilibrium distribution. One example lack of ergodicity: when the simulation fails to explore some modes in the Boltzmann distribution. Even if having explored the full state space, is possible to have off-equilibrium trajectories because of rare events, which require to be sampled "many" times to accurately represent equilibrium statistics.
Biased simulation: Simulations performed with modified versions of naive MD that speed up the exploration of state space. These methods subset of enhanced sampling and
include meta-dynamics (Laio & Parrinello 2002). replica exchange (Earl & Deem 2005) and others (Sidler et al. 2016 Pasarkar al. (023). Data generated with enhanced sam- pling do not resemble the underlying transition density, but, often, can be re-weighted to represent the equilibrium distribution.
A.2 EIGEN DECOMPOSITION OF THE TRANSFER OPERATOR AND TRANSITION DENSITY
The two-fold composition of the transfer operator, eq. 5 acting on an initial y--eiggted density, P.
20
(T((T) o (x(+2r)) = Σ (x(T)
Σ
Xs(T)olos) vslkos)
the (x++2))77
1=1
1=1
Σ Xe(T))l (05102) (x((2))))
(x(T)
( 2h (xc42r))
where we have used the orthonormality of the eigenfunctions, that is (1byl(s) deg. Similarly,
ITA (T) p|(x)+N=) = Σ (r))oll (x(x))))))
(19)
The transition density can be re-written terms of the spectral decomposition of the transfer op- erator by choosing the initial density, p. Dirac delta function dx, (x), that is p(x)
(x)
n(x)
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Then,
xx: (x(+N+) H
p(x)+NT|xx)) ((x++N+)
Th (T)
=((x+N(F) Σ (T) (T) (
6x
(i Uhs (x(++N))))
r
Σ (N(T) 0x (x)lib)) (e((Xe+NT) p(Xe+NT)
Σ (()))((())))))))))))
A.3 IMPLICIT TRANSFER OPERATOR DETAILS
Implicit Transfer Operator (ITO) Learning (Schreiner et al. 2023) is framework for learning surrogate models the transition density, p(xN, Xxo). ITO models are trained sampling tuples
(xt,) N,,,, NN and training generative model mimic the empirical transition density at mul- tiple time-horizons, Nr. Training is done following Algorithm
Algorithm Training. DisExp is defined in Algorithm 4
Input: MD-trajectories; X
{x/: x1, ) ITO score-model; év, max lag; N
X" Concatenate( (x): x 1 Nus )
while not converged do
X, (hoice(X'')
N ~ DisEExp(Nmaan
laiff ~ Uniform(O, T')
Take gradient step on:
end while (x X,, N, t2m)
Ve [||c éœ (x
faiir
return 60
Once model is trained, it can be sampled by following Algorithm 2
Algorithm 2 Sampling from Po (x N)
Input: initial condition Xo, lag: N, diffusion steps; Tait. ITO score-model; 8
Tais ~ NN 1)
XN
for Laiff = Taiff ... do
~ 1))
x
far-1
Lqin 1-a'dift Eg (xfrir Xo, N, tastt) ) + BBE
Vatai
V1 11--a
end for return x
Several sampling steps be annealed to sample longer lag-times depicted in Algorithm
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Algorithm 3 Ancestral sampling. Sampling from pe is defined in Algorithm
Input: initial condition Xo, lag N, ancestral steps n. Allocate T E ()xdmi))))((
T[o] X0
for i = 1 do
x, ~ Po (T 11,N) N)
T =x
end return T
Algorithm Sampling from DisExp
Noo Uniform(O, (0, log(Nmax) Return: floor(exp( (1og))
BoPITO uses Algorithm for training and Algorithm for sampling as well, but uses the score mode in eq. 14
A.4 DECAY TIME-DEPENDENT SCORE TERM
In Figure 6
we show the average time-dependent component of the score,
[ISM Sdyn sdyn (x Xt, N, lanr, 0)
SN E Aa
(20)
of trained BoPITO model of the Prinz Potential with Å 0.994. The expectation for correspond-
ing to the sampling of tuples as during training (Alg. []. The model remains flexible for small lags but eventually decreases to 0, sampling from the equilibrium model.
score
0.200
0.150
0.125
0.100
0.075
0.025
0.000
10a
10a
Lag
Figure 6: Average time-dependent component of the score, eq. model of the Prinz Potential.
20 of trained model BoPITO
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A.5 THE SCORE OF THE TRANSITION PROBABILITY AND BoPITO
The score of the transition probability can be written as
V XNy logp(XNr Xo, N) - Σ (i(T) (x0)
VxN- (x.N))
[D, N)
((N.H(XNT) +
- )c(T) vix(Xo)
V Vxx, pi(xNN+)
MXXNT1X0, N)
p(xN~1x00 N)
1=2
VxN,M(XNT)
(xxN==)
1((x+))
p(xN-1x00, N)
AN (x(TT)
80 N
(x(x))
p(XN, Xo, N) VxN- Fxx-di(XN-)
V xx. ((xx) is the score of the Boltzmann distribution and can be modeled with the score of a Boltzmann Generator, Seq (xc tair)) p(xN, NJ. + 1 as + oo and can be modeled with a scalar neural network + AN fe (x Xo. N, tanr). The last term can be modeled with go (x Ny. Xo. N, tair). The corresponding score model is
Pxg_u((M0 p(XNe)
m(xx))
(x Anm so(x N., xo: N, kam) Seq doq(x"t (xAmir (1 AN So(x NT: Xo, N, lair) + A go (x NT: Xo. N, lair)) S ( (xfat, fxirrr + AN (seq fail tan) fo N.. Xo. N, taa) + go (x Xu, N, fan))
sayu No N.faar
We can aggregate f and g to single neural network component, Sayu (x Ny. Xb, N. fair), to get
so Ny Xo, N, tanr) Seq Seq(x fail tain) A sdyn Ex Nr. Xu. N, fain).
(21)
For simplicity, we incorporate the structure of the transition density into the diffusion model, not only for 4an (the learned data distribution), but for all lait 0. We do not observe this choice to limit model expressivity in our experiments.
A.6 FITTING Ā
The hyperparameter controls the time-scale at which the BoPITO model transitions to sampling the equilibrium model. Ideally, Å should be similar to the eigenvalue corresponding with the slowest process the system, Ag- However, its value is not generally available and requires extensive unbiased simulation data to be accurately estimated. If À is too small, the model may prematurely relax to equilibrium, limiting its ability to capture non-equilibrium dynamics. Conversely, if À is too large, the benefits of the BoPITO framework could diminish. Therefore, careful tuning of Â can be crucial for optimal performance.
As illustrated Figure 7 grid-search hyper-parameter tuning can select an appropriate Î for single-system model. Too small values of À can hinder the model's ability to capture long-time- scale dynamics, leading to increased loss. We recommend choosing the smallest Å that yields a plateau in the À-loss curve (elhow rule), as demonstrated in Figure 7 (b) for the Prinz potential. However, practitioners should be cautious about sampling longer lags than the implied time-scale defined by Á if they cannot guarantee the system relaxes to equilibrium for those time-scales. Fitting curves for Alanine Dipeptide and Chignolin are shown in Figure g
A.7 ALTERNATIVE SCORE FORMULATIONS
The score in eq. 14 is principled and resembles the score of the transition density. However, in practice, found that this score lead to suboptimal performance in learning short time-scales the dynamic component fails to dominate the equilibrium component for small N. We only observed this for Chignolin. We relate this issue numerical limitations due very different scale
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10-1
0.92
0,4300
0.94
x
0.96
0.4295
0.98
Loss
4.36x10
0.99
Loss
0.4290
x0
0.996
4.32
0.4285
40-x
0.4280
4.28 10-1
0.4275
10
102
0.90
0.92
0.94
0.96
0.98
1.00
Epoch
Á
(a)
(b)
Figure 7: Average loss against training epochs for the Prinz potential for different values of İ (a) and average final loss versus Ả (b). Averages taken w.r.. 10 runs. Final losses computed the average between epochs 190 and 200. The greatest eigenvalue of the system under our simulation parameters is 0.994.
0.250
0.31
0.245
0.30
Loss
Loss
0.240
0.29
0.235
0.230
0.28
0.225
0.9900
0.9925 0.9950 0.9975 1.0000
0.9900.99229999449996..999
(a)
(b)
Figure 8: Average final loss against different values of À for Alanine Dipeptide (a) and Chignolin (b). The smallest value of À that allowed numerically stable training for Chignolin was 0.9995. Orange cross symbolizes that the trainings crashed because of numerical instability reasons.
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in neural network weights for different values of N. We remark that short time-scales are the least relevant since they can be easily sampled with MD. Still, to mitigate this effect we propose the following alternative factorization,
se s (X NN,
"" tam)) + \ Sdyn (x fall N, tdit)) = f(N) (x"""r
Nr No, N, tdift),
(22)
where f(N) is an increasing function and tends to 1 for large N. One potential option that does not introduce extra hyper-panee pareer-araetters is
A-2N \2N
NT, Xo: N, Edift) --2N X2N Seq tan (ditt) X sdyn (XNr Xo, N, fair).
(23)
Moreover, to ensure adequate convergence to the equilibrium distribution and avoid over-fitting to unbiased MD for long time-scales, we use the following decay score for N > Ndecay.
se (x Xo, .N, tain) = (xq taitt) + X sdyn (xx tain
Nr Xo, NNoccay tait))
(24)
We use this version for the experiments conducted on Chignolin and set Nawcy 3/2, where 12 is the unitless characteristic time of the slowest process in the system, 12 log
-1
A.8 How BOPITO INTERPOLATORS MITIGATE OUT-OF- OUT-OF-DISTRIBUTTIN DISTRRIBTTTUI EFFECTS
BoPITO interpolators mitigate citt---ddistiibttinn effects by, first, fixing N = Nmar the argument of 6dyn This choice ensures that none of our neural networks are evaluated outside of the training domain. Second, alter long-lag interpolation steps with short-lag non-interpolation steps for local relaxation. Since we in practice see that that the long-lag steps occasionally generates 'off data manifold" states, e.g. structures with potential high energies, the local 'relaxation' steps projects us back onto the data manifold. In practice, just one short-lag non-interpolation step is sufficient overcome this. Third, we use dynamic observables to fit the interpolation parameter. Without any additional source of information we don't know how to choose Nine. However, when we use dynamic observables, we can select the interpolated ensemble which is the most consistent with experimental observables. So this helps calibrate the time-scales to stay "in distribution".
A.9 EXPERIMENTAL PARAMETERS
A.9.1 BOLTZMANN PRIORS FOR DATASET GENERATION
For different numbers of trajectories, n, we train 5000/n ITO models on n Prinz potential trajecto- ries of length 150. Trajectories do not overlap among different trainings. The maximum model lag 100 steps.
A.9.2 BOPITO SAMPLES EFFICIENTLY LONG-TERM DYNAMICS IN DATA-SPARSE SCENARIOS
Prinz potential 10 models are trained on non-overlapping trajectory sets for different numbers of trajectories. The trajectories length is 10, 000 and the maximum model lag is 1, 000. We consider lags of 10, 25, and 50 as short, 100, 200, and 300 as medium, and 500, 750, and 1, 000 as long time-
scales in our experiments, see Figure 10 (b) for a visual reference. À is set to 0.994 for all BoPITO experiments.
Alanine Dipeptide 10 models are trained on potentially overlapping random trajectory sets for different numbers of trajectories. The trajectories length is 12, 500 and the maximum model lag is 10, 000. We consider lags of 5, 10, and 50 as short, 100, 500, and 1000, as medium, and 2500, 7500, and 10000 as long time-scales in our experiments, see Figure 11 (b). Á is set 0.9996 for all BoPITO models.
Chignolin 4 models are trained on potentially overlapping random trajectory sets for differ- ent numbers of trajectories. The trajectories length 35, 000 and the maximum model lag 30, 000. We consider lags of 10, 50, and 100 as short, 500, 1000, and 5000, medium, and 10000, 20000, and 30000 long time-scales in our experiments, see Figure 13 (b). Â is set 0.9995 for all BoPITO models. BoPITO models are trained using the score in eq. 23 and the decay function in eq.
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Diffusion steps
500
Diffusion steps
1. 000
Noise schedule
Sigmoidal
Noise schedule
Polynomial
Batch size
2, 097, 152
Batch size
1, 024/32
Learning rate
0.001
Learning rate
0.001
Layers
3
Score layers
5
Embedding dimension
256
Embedding layers
2
Net dimension
256
nefeuueres
64
Optimizer
Adam
Optimizer
Adam
Inference ODE steps
50
Inference ODE steps
100 /50
Table 1: Architectural and training parameters of MB-ITO models (a) and SE3-ITO (b). Batch sizes
and inference ODE steps refer Alanine Dipeptide and Chignolin experiments respectively.
A.9.3 INTERPOLATING BETWEEN MODELS TRAINED OFF EQUILIBRIUM AND BOLTZMANN DISTRIBUTION
We remove the transitions resembling the slowest process in the system, 6 crossing 0 or 2, gen- erate biased simulation data. When transition occurs, we remove one frame before and after the transition and split the trajectory. We train both biased ITO and BoPITO models with maximum model lag of 100 on the resulting biased dataset. For lags >> 100, we sample the BoPITO interpolator fitted the unbiased dynamic observable defined by eq. 26 Figure We perform interpolation by sampling with the score model in eq. [5 followed by one round of non-interpolation sampling with lag 100 steps for local relaxation.
A.9.4 ARCHITECTURAL, TRAINING AND INFERENCE DETAILS
We use MB-ITO as the architecture of models for the Prinz potential, and SE3-ITO, an SE(3) equivariant neural network, for Alanine Dipeptide, both introduced in SSchreinrr et al. 2023 We report architectural and training hyper-parameters in Table [ Models are trained until convergence in the log-log loss plot.
A.10
PRINZ POTENTIAL
The Prinz potential is ID potential commonly used for benchmarking MD) sampling methods. The potential is defined as
U(x) 4 (22 0.8e 80x² + -.22-80(( 0055) 0.5c 40(=+)055)'
(25)
We generate trajectories using an Suler-Maruyami integrator using the library Deeptime (Hoffmann et al. (021). We set the integrator time-step to 1 10 5 and the temperature, mass, and damping factor to In Figure we show histograms of the position of a particle after steps, starting from 0.75, and in Figure 10 (a) we report an implied time-scales plot. We use the identity function to define a dynamic observable/corrrlation function of this system and show it in (b).
A.11 ALANINE DIPEPTIDE
Alanine Dipeptide is a small peptide with 22 atoms. We use publicly available data from Dibak et al. (2022), containing total of 1 μas simulation time split in 20 trajectories. Simulation is performed in an implicit solvent with 2 fs integration time-step and data is saved every ps. We choose,
al b(x)) = sin Φ
(26)
cos Φ
to define dynamic observable of this system. The torsion angle ø is involved in the slowest tran- sition observed in the simulation, see Figure 12 We combine these vectors taking the inner product
for computing dynamic observables. We show this correlation function in Figure (a), and implied time-scales plot in Figure (b) and a histogram the torsions and & aggregating all simulation data in Figure T2
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Log-10
Lag-25
Lag-50
1.5
2.0
1.5
1.5
1.00
0.5
0.5
0.0
-0.5
0.0
0.5
1.0
0.0
-1.0
-0.5
0.0
1.0
0.0
-1.0
-0.5
0.0
0.5
Lag-100 100
500
Lag-1000
1.5
Kaler- Mallr-Marugamia
Boltamano
0.8
1.0
0.6
0.5
0.4
0.4
0.2
0.2
0.0
0.0
0.0
-0.5
Figure 9: Conditional density of a simulation of the Prinz potential starting at r 0.75 for different lags (steps). Long-term dynamics approaches the Boltzmann distribution.
Timescale (steps)
Correlation
0.6
10a
0.4
0.0
10
15
20
101
10a
10a
Lag time (steps)
Lag
(a)
(b)
Figure 10: Implied time-scales (a) and dynamic observable under the identity function (b) for the Prinz Potential. Errors in (b) are smaller than the dots. In (a), the implied time-scales of the slowest processes in the system are computed for different lags. The color order. from longest to shortest implied time-scale, is blue, orange, green, red, and purple
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0.4
Timescale (steps)
Correlation
0.3
0.2
104
0.1
0.0
0
20
40
60
80
100
10a
10a
10a
10a
10a
Lag time (steps)
Lag time
(a)
(b)
Figure 11: Implied time-scales for Alanine Dipeptide computed with Markov State Model torsional angles & and v (a) and dynamic observable in eq. (b). In (a), the implied time-scales of the 5 slowest processes in the system are computed for different lags. The color order, from longest to shortest implied time-scale, is blue, orange, green, red, and purple.
-3
-2
0
Figure 12: Histogram of the torsional angles o and of Alanine Dipeptide (insert). Data is aggre- gated among all trajectories in the dataset introduced Dibak et al. (2022).
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1.0
1.00
0.8
0.75
Foldedness
Correlation
0.6
0.50
0.4
0.25
0.2
0.00
0.0
10000
20000 Frame
30000
10a
10a
10a
Lag
(b)
Figure 13: Time evoluation of 'foldedness" in a subset of the reference MD simulation (a) and dynamic observable in eq. 27 (b).
A.12 CHIGNOLIN
Chignolin (cln025) is a fast folding protein with 10 residues, 166 atoms and 93 heavy atoms. We molecular dynamics data previously reported by Lindorff-LLasen al. (2011). The data is proprietary but available upon request for research purposes. The simulations were performed in explicit solvent with 2.5 time-step and the positions was saved 200 ps intervals. We extract all heavy atoms positions from the simulations and train models on this data. We use the reaction coordinate proposed in Lindorff-Larsen al. (201T) as define dynamic observable,
Ei
Ni N; jii
1tc (x)
a(x) b(x)
(27)
Σ Mo 2i=1 N
where the first in the numerator iterates over all Nres residues in the protein, while the second sum considers the N; native contacts of residue separated by at least seven residues in the primary sequence. We define native contacts as residue pairs separated by at least seven residues in the primary sequence and with Ca atoms closer than 10 Á in the native structure. Moreover, dy (x) and di represent the distances between the C., atoms of residues and in the structure and the native structure, respectively. This observable quantifies the protein's 's foldedness, with values ranging from (unfolded) to (folded). In Figure [3] (a) we show the evolution of eq. [2 on subset of the reference simulation data and we observe how the protein undergoes transformations between the folded and unfolded states. In Figure 13 (b) we visualize the corresponding dynamic observable.
A.13 METRICS
We evaluate models computing differences in the dynamic observables introduced in section 5.1 w.r.t. long MD simulations,
Acorrelation) N O0 Omodel|
(28)
where O is the normalized observable predicted by MD for lag N and Ormadel is the model's prediction. See Appendix A.15 for details on normalization. We report the average difference over different training runs.
A. 14 ENERGIES FOR ALANINE DIPEPTIDE
In Figure 14 we observe a remarkable agreement in energy densities of samples generated with our Boltzmann Generator and MD. GBBAOBoorce is the implicit solvent model energy.
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Published as a conference paper at ICLR 2025
orc
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Energy [kg
Figure 14: Energy and energy components of samples generated with Boltzmann Generator (BG)
and MD.
A.15 CORRELATION FUNCTION NORMALIZATION
We subtract the mean and divide by correlation at lag 0 to normalize our dynamic observables. Subtracting the mean guarantees that the correlation function asymptotically decays to and divide by the correlation at lag 0 so that correlation can be read as fraction of correlation at time The resulting normalized correlation function
El(f(x) (x)(),)) (g(x++0)) B(g(x(+000)
(29)
(x( (x[((x))) (g(x,) ((g((x)))
When comparing different methods, E[(f(x,) Elf(x,)) (g(x,) (x,()) using long MD simulations and use these same normalizing factors for all methods.
we compute [[f(x:)). Elg(x++ar) and
29


Reviews:
[{"review_id": "5AYR4gYE7f", "reviewer": "Reviewer_r47p", "summary": "This work looks at predicting molecular dynamics simulations quickly over time, using longer time steps. The authors introduce a method using diffusion models to train on both unbiased and biased MD data. The diffusion model is used to estimate the transition density, with the goal of simulating MD more quickly. BoPITO, using Boltzmann priors, is supposed to be more sample efficient than ITO (Implicit Transfer Operator). In BoPITO, the authors also use pre-trained models of the equilibrium distribution to improve the ITO method. The authors test the method on the 1D Prinz potential and alanine dipeptide.", "questions": "- I do not fully follow the advantage of BoPITO recovering the unbiased Boltzmann distribution over long time horizons: isn’t this something that only applies to equilibrium distributions? But at the same time, it seems like the authors are saying they can incorporate off-equilibrium data and/or estimate something about off-equilibrium simulations?\n\n- Can the authors compare their method to other machine learning methods for molecular dynamics simulation?\n\n- Is energy being conserved? How can you guarantee physical dynamics?\n\n- Can the authors show that this method has potential to be broadly applicable by seeing if it works for harder systems, such as larger proteins, or polymers?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- This method could enable running MD simulations over much longer timescales\n- By incorporating Boltzmann priors into the training process, this could allow more data efficiency and also ensure recovering the Boltzmann distribution over time", "weaknesses": "- There are many ways to do molecular dynamics simulations. A common way is through methods like machine learning potentials. While the time steps that are taken there are smaller, these potentials are much more accurate and have been used for more diverse systems. It would be helpful if the authors provided some comparison to this method, as well as other methods that also try to directly predict molecular dynamics.\n\n- The authors have only run this on two very toy systems: the 1D Prinz potential and alanine dipeptide. These systems are very far off from getting closer towards practical utility of this method. It would be more convincing if there were other systems studied, such as larger proteins or systems like the polymers or electrolytes system from Fu et al. (2023) TMLR. \n\n\n- This method does not seem to guarantee important properties such as energy conservation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tS8Pvw1OHi", "reviewer": "Reviewer_4QuU", "summary": "The authors point out that a prior work Implicit Transfer Operator (ITO) relies on extensive unbiased simulation, and suggest Boltzmann priors to improve the ability to approximate dynamics. To be specific, the authors first train a score model using the equilibrium data, afterwards fix the score model and only train the time-dependent components with unbiased and possibly off-equilibrium MD data.", "questions": "1. Definition regarding bias and off-equilibrium\n\nThe papers uses the term biased & unbiased, off-equilibrium. As far as I understood, the definitions could be summarized as the following:\n- Off-equilibrium data: simulation results (trajectories) that did not explore some modes of Boltzmann distribution\n- Biased simulation & samples: simulation done & sampled achieved by biased molecular dynamics, e.g. enhanced sampling, a modified version of the naive molecular dynamics\n- Unbiased simulations & samples: simulation done & sampled achieved by naive molecular dynamics, possibly may not find some modes if the time horizon is too short\n\nTo sum up, is the following a correct summarization for the method in section 3.2 and section 4?\n- BG are trained with unbiased and equilibrium data for a score model\n- Then the time-dependent components score are then trained with unbiased data, possibly off-equilibrium \n- Section 4.3 presents that BoPITO can approximate long-term dynamics in the above situation\n- Section 4.4 shows that the training data for BG are biased, and BoPITO can recover it?\n\n2. Boltzmann generators in prior\n\nBoltzmann generators [1] were trained by combining two modes: training by energy and training by example.\n- Training by energy: Generate a distribution in the configuration space from a Gaussian prior using the model, and minimize the difference between Boltzmann distribution using the energy term as weights.\n- Training by example: Since training by energy alone tends to focus on sampling on the most stable meta-stable state, they additionally use training by some valid configurations, e.g., trajectories from naive MD simulations.\nNow, my question is about section 4.2 of the paper, where “BG is available before data collection” and “BG trained on equilibrium data” seems to be a contradiction. I may have misunderstood something, could the authors please clarify this part?\n\n3. Derivation of parameterization\n\nIn Appendix A.3, the authors derive the score of the transition probability. When deriving equation, is $s_{dyn}(\\ldots) = s_{eq}(x^{t_{diff}}, t_{diff}) f_\\theta(\\ldots) + g_\\theta(\\ldots)$ ?\n\n4. Comparison to ITO\n\nFrom Figure 3 to Figure 4, the authors compare ITO and BoPITO. As far as I understood, the difference between these two methods is the parameterized epsilon, i.e. score function. They have separated the equilibrium part and time-dependent components, and have additionally trained the score function by fixing the equilibrium part and only updating the time-dependent term. Are all the other components, i.e., model and data, identical?\n\n[1] Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning, Science 2019", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper applies Boltzmann generators in ITO models to capture modes that have very small probability mass in the Boltzmann distributions.", "weaknesses": "1. Pre-trained Boltzmann generator (BG)\n\nMy main concern with this paper is that it assumes a pre-trained BG. Since BG is trained using the energy function and data samples (more detail in Q2), I am a bit confused about what the authors intended. Is BG only used to generate initial conditions for simulations, for various sampling across the configuration space? \n\n2. Lack of detail of less extensive use of data compared to ITO\n\nOne of the important limitations the authors pointed out was the need for extensive,  unbiased molecular dynamics data. However, I do not see any results from the experiment nor analysis that BoPITO succeeds in generating results similar to ITO with less data, especially for Alanine Dipeptide.\n\n3. Limited baselines and molecular systems\n    - While the authors claim that ITO can be improved with BG, however, I don’t see enough experimental results to conclude it.\n    - Comparison with ITO for molecular systems is done for only Alanine Dipeptide, while ITO has presented qualitative results and quantitative results for the folding process of Chignolin.\n    - Comparison of quantitative stationary and dynamic observables to naive molecular dynamics has not been presented, while it has been for ITO for four molecular systems (Chignolin, Trp-Cage, BBA, Villin). To support the claim that BopITO truly predicts time-correlation statistics (dynamic observable), this seems to be needed.\n    \n    Overall, it seems that the paper does not present enough experiment results to conclude that BoPITO improves compared to ITO [1].\n    \n4. Missing ground truth transition probability for Alanine Dipeptide\n\nOne main and important result is the condition transition probability density for Alanine Dipeptide, figure 5. Though the figure shows that BoPITO and ITO (unbiased) show similar results, there are no ground truth data given to compare with. Though I understand it might be impractical to compute the transition densities for alanine dipeptide for 15,000 trajectories of length up to 1 nano-second compared to trajectories of length 500ps as in ITO[1], it would be good if the authors clarified this.\n\n5. Requiring a well-trained Boltzmann generator\n\nSince the proposed method uses Boltzmann generators as priors, a well-trained Boltzmann generator is essential. Although it has rather easily been done for Alanine Dipeptide, it seems that extension to complex molecular systems such as Chignolin is not flexible. \n\n[1] Implicit Transfer Operator Learning: Multiple time-resolution surrogates for Molecular dynamics, NeurIPS 2023\n\nMinor\n\n- Typos\n    - Page 4 last paragraph, improve ITO learning in three ways ⇒ fourth ways\n- Some notations are not clarified & used\n    - $x$: molecular configuration first in page 1, it is the 3D coordinates of molecular system being $x \\in \\mathbb{R}^{3 \\times M}$ where $M$ is the number of particles?\n    - Diffeomorphic map $\\mathcal{F}_\\theta: \\mathbb{R}^N \\rightarrow \\mathbb{R}^N$ , are the dimension of the latent space and configuration space necessarily identical?\n- section 2.3 - transfer operators\n    - The part explaining equation 5, the eigenvalues, and right and left eigenfunctions sounds awkward.\n- Figure 3\n    - There are no (a), (b), (c), (d) in the figure, as it is pointed out in section 4.3.\n    - The authors ought to show that using BG as priors is better in terms of different time scales and number of trajectories, so it might have been better to plot the columns and legends switched, e.g., ITO / BoPITO and different time scales.\n- Figure 5\n    - I can’t find the black cross in the first column indicating the initial condition. Could you increase the size of the mark or change the color to something like red?\n- Figure 8, 9\n    - Explanation on the colors is missing. Could you add a legend for it?\n- Equation 15\n    - Theta seems to be missing in ${s_{dyn}}$\n    - Subscripts for ${x^{t_{diff}}}$ seems to be missing?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8b2xSRCXVz", "reviewer": "Reviewer_nG3C", "summary": "This paper proposes to combine Boltzmann Generators (BG) with Implicit Transfer Operators (ITOs). ITOs are limited because a lot of unbiased molecular dynamics (MD) data is needed for training. This framework combines ITOs with a BG before being able to sample efficiently and be asymptotically unbiased from the equilibrium distribution. \n\nFrom a more practical standpoint, the score function of the Score-based Diffusion Model is separated into an equilibrium contribution (from the BG prior) and the time-dependent component (to tackle off-equilibrium simulations). BoPITO defines an interpolation between models trained on off-equilibrium simulations and the equilibrium distribution, thus making it a more efficient sampler of long-time MD simulations.", "questions": "- At the bottom of page 2 the authors write: \"As a result, most MD data will be ‘off-equilibrium’, e.g. trajectories exploring one or a few of the modes of $\\mu(x)$\". Is it correct to interpret 'off-equilibirum' in this case as 'non-ergodic'?\n- In line 59-60: \" [...] BoPITO, by construction, guarantees asymptotically unbiased equilibrium statistics.\". Where do asymptotic guarantees for unbiasedness come from? I presume that reweighting still needs to be enforced in order to have an asymptotically unbiased sampler [1] though it is not immediately clear to me where this occurs. \n- in line 116-117: \"BGs are trained either with approximately equilibrated simulation data or with biased simulations, from i.e. enhanced sampling, by employing appropriate reweighting[...]\". I think this is not entirely correct, strictly speaking. BGs can also be trained in a data-free manner and a combination of data-informed (biased/unbiased) and data-free losses (see [2]). Did the authors solely rely on negative log-like hood training or also on data-free losses (e.g., Reverse KL)?\n- I believe it would be useful to have a reference around line 120 about sampling with reweighting to enforce asymptotic unbiasedness, see e.g., [1]. \n- What do the authors mean by **importance re-sampling**?\n- When looking at the results in Fig. 2 I am not sure what is the crystal baseline. Is it some SOTA or how is this obtained?\n- As a follow-up, question: are there any other ML-based models that can be used in this case in order to prove the goodness of the proposed method? e.g., some other baselines that are worth comparing here such as naive ITOs, or some sort of BGs able to handle MD at different time steps [3]. \n- Is there an intuitive explanation for having in the top right corner of Fig. 3 the long time scale difference in correlation to be smaller than medium and short?\n- What is an intuitive understanding of $N_{\\textrm{int}}$? Is that the resolution in the interpolation process between equilibrium and off-equilibrium distribution? In other words, some sort of annealing parameter?\n- Are there any baselines to compare against in Fig.5? See also the weakness stated above in terms of lack of comparison with existing methods. For instance, does FAB [4] allow the computation of the same type of dynamics? If yes, how does it compare to BoPITO? If not, why?\n- This is more of a general detour question due to my limited knowledge of MD simulations. In line 475 onwards, the authors write: \"These approaches, in general, aim to accelerate molecular simulations akin to BoPITO, yet due to the CG operation, the molecular dynamics (kinetics) will be accelerated, and detailed knowledge of the unbiased dynamics are needed to correct this [...]\". I am wondering how this can be possible. When coarse-graining a system and running MD at a coarser level, you inevitably lose information on the degrees of freedom at a smaller scale. I am naively wondering if there are ways to rigorously correct for this and still retain asymptotic guarantees in this case. \n\n- Lastly, I am wondering if the following intuition behind this work is correct: I train a core component of the equilibrium distribution using BGs. Then, I use some biased MD data to refine my backbone model in order to capture rare events, making it an 'ergodic sampler'. The idea is to use the ITOs paradigm to have some smooth interpolation between the equilibrium-off-equilibrium regimes encoded in the model in different ways (BG+biased MD data).\n\n\n## References\n- [1] [Nicoli, Kim A., et al. \"Asymptotically unbiased estimation of physical observables with neural samplers.\" Physical Review E 101.2 (2020): 023304.](https://link.aps.org/accepted/10.1103/PhysRevE.101.023304)\n- [2] [Noé, Frank, et al. \"Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning.\" Science 365.6457 (2019): eaaw1147.](https://www.science.org/doi/10.1126/science.aaw1147)\n- [3] [Klein, Leon, et al. \"Timewarp: Transferable acceleration of molecular dynamics by learning time-coarsened dynamics.\" Advances in Neural Information Processing Systems 36 (2024).](https://proceedings.neurips.cc/paper_files/paper/2023/file/a598c367280f9054434fdcc227ce4d38-Paper-Conference.pdf)\n- [4] [Midgley, Laurence Illing, et al. \"Flow annealed importance sampling bootstrap.\" arXiv preprint arXiv:2208.01893 (2022).](https://arxiv.org/pdf/2208.01893)", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The paper is clearly written and easy to follow. Overall, it represents a pleasant read. \n- The authors made a great effort to make the paper self-contained, including a lot of background sections covering the basics of MD, BGs, ITOs, and diffusion models. \n- Numerical results are convincing in showing advantages compared to ITOs (PRO), although the approach is not compared to other baselines/methods in the literature (CON). This allows us to only partially appreciate the advantages of the algorithm. \n- The authors address the limitations of the current methods and envision possible avenues for future work. \n- The code attached as additional material looks good, properly formatted, and seems to run out-of-the-box", "weaknesses": "- I find the main weakness of this work being the lack of comparison with other ML-based approaches that can deal with sampling at different time scales (e.g., [3,4]). Moreover, it would also be useful to show a ground truth in the plots when possible, e.g., Fig. 5. Adding more baselines would certainly improve the manuscript. \n- As a suggestion, rather than a weakness, I feel it would be better to move the discussion to the related work at the beginning of the paper. This would help people to associate some works they may already know with the present work in case they are familiar with the literature. This would represent a useful prior to have before getting into the core of the paper. \n- At the bottom of page 4, the authors say that BoPITO improves ITO learning in three different ways. However, in what follows, there are four points. I also believe it would be helpful to list these contributions visually with bullet points.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pJWN9ZBRoS", "reviewer": "Reviewer_hu2r", "summary": "The paper uses a pretrained Boltzmann Generator to include un-biased equilibrium information into Implicit Transfer Operators (ITOs).\nITOs learn \"time coarse grained\" transition densities from molecular dynamics (MD) data.\nThe challenge is that ITOs require (a) unbiased (b) large quanities of MD data to train.\nBoltzmann Generators (BG) can approximate the true Boltzmann distribution even if trained with biased data when combined with importance resampling.\n\n\nThe paper suggest three ways to improve ITOs with Boltzmann generators:\n1. Sampling of conformations proportional to the equilibrium distribution from the Boltzmann Generator as starting points for trajectories\n2. Inductive bias: Separate learning of the transition density into a fixed stationary part (the Boltzmann generator) and learned time-dependent component to improve sample efficiency.\n(Based on the spectral decomposition of the transfer operator, where the equilibrium distribution is an eigenfunction with an eigenvalue of one.) An explicit temporal decay of the time-dependent component ensures convergence to the unbiased Boltzmann distribution for long-time horizons.\n3. Approximate rare dynamics outside the data, by interpolating between ITOs trained on off-equilibrium data (time-dependent component) and the unbiased equilibrium distribution (Boltzmann generator).", "questions": "* The formalism decomposes the operator in a sum of left and right eigenfunctions. Are these eigenfunctions used anywhere in the training objectives directly, or are they solely used to justify/derive the form of the score function ansatz? \n* In line 139 and 142 it says $\\lambda_i = exp(-\\tau \\kappa_i)$ but also that $-1 < \\lambda_i < 1$. How is it possible that $\\lambda_i = exp(-\\tau \\kappa_i) < 0$ though?\n* I am having some trouble understanding the interpolators section. N_max is the maximum training lag, but then we sample with N_int > N_max. Doesn't that cause out-of-distribution problems? Maybe you could add a bit more explanation in the appendix as to why the given model implies an interpolator. \n* line 60: How exactly do you define \"off-equilibrium\" and \"biased\"? \n* line 191 and following: what is $t_{diff}$?\n* Looking at Figure 3, it seems like ITO and BoPITO have very similar performance on short and medium time scales, and BoPITO mostly outperforms ITO on long timescales, presumably due to the enforcement of the Boltzman distribution for long time horizons. I am wondering how often it is precisely the medium time scales that we are interested in though? \n\n## Suggestions\n* Include a section that explains the ITO framework in more detail and includes pseudo-code\n* line 107 and other places, I find formulations like \"few of the modes of μ(x)\" are easier to read if written as  \"few of the modes of the *Boltzmann distribution* μ(x)\"\n* line 209 says \"improve ITO learning in three ways\" but then four points are listed", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "* Temporal coarse-graining of molecular dynamics is of very high relevance to simulate long-horizon observables more efficiently\n* Making use of cheaply obtainable off-equilibrium data is a very promising approach to train more scalable models\n* Method seems sound and well-founded in physical/statistical principles \n* The results look promising, although somewhat limited due to the small scale experiments", "weaknesses": "* The paper would benefit from clearer writing and more extensive background sections. It seems to assume that readers are already familiar with the implicit transfer operators framework, which will often not be the case. More explanations and pseudocode (even if it is just in the appendix) would make the paper much easier to read.  \n* The experiments are conducted only on small-scale systems, and it is unclear how well this approach would scale to more realistic scenarios.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)