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Machine learning kinetics from molecular dynamics data

Jonathan Weare, Aaron R. Dinner

arXiv:2609.17736v1physics.comp-phcond-mat.stat-mechcs.LGphysics.chem-phphysics.data-an

TL;DR

Molecular transitions often occur beyond direct MD timescales, and TPT does not prescribe how to estimate committors from simulation data. This review connects equation-based, variational, trajectory-based, and neural-network approaches, concluding that direct estimation from trajectories can extend to settings where the dynamics are unknown.

  • Problem

    TPT uses committors and stationary distributions to compute rates and reactive fluxes but does not prescribe how to estimate those quantities from MD data.

    Source: B. Roadmap · p. 2

  • Method

    The review connects generator-based PDEs and neural networks with trajectory-based kinetic-statistics methods, emphasizing direct estimation from trajectory data.

    Source: B. Roadmap · p. 2, §VII · p. 16

  • Results

    The review concludes that direct trajectory-based estimation extends naturally to reduced variables, irreversible dynamics, and experimental time series where the dynamics’ form is unknown.

    Source: §VII · p. 16

  • Takeaways & Limitations

    Stopping trajectories at metastable-state boundaries and achieving good transition-region sampling are key technical considerations, and committor estimates can guide sampling.

    Source: §VII · p. 16

    Generator-based methods require explicit evaluation of L, while reduced-coordinate dynamics may have an unavailable projected diffusion tensor.

    Source: B. Variational rate estimation · p. 4

Abstract

from arXiv · show

Most molecular transitions occur on timescales far beyond direct molecular dynamics simulations. The committor, the probability that a configuration reaches a product state before a reactant state, is a central kinetic statistic, providing a mechanism-independent reaction coordinate and a foundation for transition path theory and the calculation of rates. This review surveys modern approaches for estimating the committor and related kinetic statistics from molecular simulations, with an emphasis on self-supervised methods that learn solutions of their defining dynamical equations rather than relying on labeled shooting data. We develop a common operator viewpoint connecting generator-based partial differential equations, variational principles, Markov state models, dynamical Galerkin approximation, and neural networks. Empirical and theoretical evidence points to the efficiency of these methods. We provide theoretical and practical guidance for realizing their full potential in applications, including strategies for treating non-Markovian effects and for sampling. We conclude by identifying opportunities for further research, including connections to reinforcement learning and generative modeling.

I. INTRODUCTION

Molecular dynamics cannot directly reach many experimental timescales, while identifying useful progress variables is difficult and conventional rate theories can miss transitions with multiple diffuse pathways. This review focuses on estimating the committor from simulation data, connecting equation-based approaches and neural networks to kinetic statistics and sampling.

I. INTRODUCTION

  • MD timescales fall short of many experimental timescales, making unbiased transition events prohibitively costly to observe with adequate statistics.The review gives ∼10−5–10−4 s for atomic-resolution simulations and ∼10−3–104 s for the molecular examples discussed.Source: §I · p. 1

A. Rate theories

  • Transition state theory generally overestimates rates when hypersurface recrossings occur, while Kramers-type theories assume a single well-defined saddlepoint that many modern systems lack.Some transitions instead involve competing pathways and broad, diffuse transition-state regions.Source: A. Rate theories · pp. 1–2

B. Roadmap

  • TPT provides a framework for rates and reactive fluxes from committors and the stationary distribution, but not a prescription for estimating them from MD data.The review therefore focuses on estimating the committor, whose value is the probability of reaching B before A and whose q = 1/2 level identifies transition states without prior mechanism knowledge.Source: §I · p. 1, B. Roadmap · p. 2
  • The review contrasts supervised regression on shooting outcomes with self-supervised methods that fit defining equations and boundary conditions without independent MD simulations.It connects generator-based PDEs, MSMs, DGA, and neural networks, and identifies sampling and departures from the Markov approximation as topics for later discussion.Source: B. Roadmap · pp. 2–3

II. PDE APPROXIMATION OF THE COMMITTOR

The committor and related kinetic statistics can be expressed through equations of the dynamics’ generator. These equations connect trajectory probabilities and rewards to the quantities used to characterize transitions.

II. PDE APPROXIMATION OF THE COMMITTOR

  • The committor is the probability of reaching B before A, and its flux equation follows from the backward Kolmogorov equation for conditional probabilities.For the committor, trajectories stop on A ∪ B with zero running reward and terminal reward 1 in B and 0 elsewhere.Source: §II · p. 3
  • The generator L acts on functions of the initial state x, and its expectation-based definition describes the dynamics’ infinitesimal time evolution.The generator includes the Liouville operator when the dynamics have a deterministic component.Source: §II · p. 3
  • The same reward formulation gives the MFPT by stopping at B, accumulating unit running reward, and assigning zero terminal reward.The MFPT is zero in B; more generally, the approach extends to conditional expectations.Source: §II · p. 3

A. Physics-informed NNs (PINNs)

PINNs approximate the committor by minimizing a generator-equation residual while enforcing boundary conditions. Variational methods provide an alternative for reversible dynamics, with practical constraints on generator evaluation and sampling.

A. Physics-informed NNs (PINNs)

  • PINNs represent trial committors with parameterized functions and minimize the PDE residual using samples from a user-chosen distribution.Boundary conditions can be added as a loss term or imposed by masking; masking satisfies them exactly but can create discontinuities near A and B.Source: A. Physics-informed NNs (PINNs) · pp. 3–4
  • Second-order derivatives make PINN training numerically challenging, require smooth neural-network activations, and can make cost scale quadratically with system dimension.These issues motivate the other methods reviewed in the paper.Source: A. Physics-informed NNs (PINNs) · p. 4

B. Variational rate estimation

  • For reversible dynamics, the committor minimizes the Dirichlet form under its boundary conditions, and trial functions yield rate overestimates.The rate expression requires the generator explicitly; reduced-coordinate applications may lack the projected diffusion tensor, and assuming it constant and isotropic is generally inaccurate.Source: B. Variational rate estimation · p. 4
  • The Dirichlet form needs only first derivatives but requires stationary-distribution sampling and is restricted to reversible dynamics.For nonequilibrium systems, the review points to PINN or trajectory-based methods instead.Source: B. Variational rate estimation · p. 4

III. LEARNING THE COMMITTOR FROM TRAJECTORY DATA

Trajectory-based methods estimate kinetic statistics from saved data without an explicit dynamics model. MSMs solve for a discretized committor, but retain discretization and finite-lag errors even with unlimited data.

III. LEARNING THE COMMITTOR FROM TRAJECTORY DATA

  • Trajectory-based approaches compute committors and other kinetic statistics from finite-interval data without explicitly specifying the dynamics.They are useful when generator evaluation is unavailable or undesirable, including reduced-coordinate analyses, black-box simulations, and experimental data.Source: §III · p. 4

A. Markov state models

  • MSMs discretize dynamical variables into states and estimate transition probabilities, then solve a linear system for statewise committor values with boundary values on A and B.The transition matrix is an operator expansion in indicator functions, and the sample distribution can be chosen freely because statistics are conditioned on the initial state.Source: A. Markov state models · pp. 4–5
  • Even with infinite data, MSM committors retain state-discretization error and a finite-lag-dependent bias.Fine discretization near rapidly changing committor values can mitigate the first error but spreads finite data across more states; continuous representations address discretization, while an exact-statistics operator addresses finite-lag error.Source: A. Markov state models · p. 5

B. The stopped transition operator

Stopping trajectories at terminal states makes the finite-lag committor equations exact, while dynamical Galerkin approximation (DGA) solves them from sampled trajectory data using a chosen basis. DGA accommodates user-chosen sampling distributions and includes stopped Markov state models as a special case.

B. The stopped transition operator

  • Stopping trajectories at A or B, or at lag time τ if earlier, avoids the finite-τ bias of unstopped equations.The stopped equations remain exact for finite τ, allowing τ to be long enough for effectively Markovian dynamics.Source: B. The stopped transition operator · p. 5

C. Dynamical Galerkin approximation

  • DGA approximates a statistic such as the committor by a basis expansion and determines its coefficients through a linear system.The basis functions vanish on A and B except for the term that accounts for boundary conditions.Source: C. Dynamical Galerkin approximation · p. 5
  • DGA estimates the system from trajectory segments started under a user-chosen distribution, whose form need not be known.Segments stop at τ or upon reaching A∪B; the estimated coefficients then define the statistic.Source: C. Dynamical Galerkin approximation · p. 5
  • With indicator functions, DGA reduces to a stopped Markov state model; alternative bases can reduce discretization error.Reported choices include boundary-vanishing pairwise-distance features, polynomial, Fourier, and radial basis functions, and learned bases.Source: C. Dynamical Galerkin approximation · pp. 5–6

1. Applications

The applications illustrate how DGA-based committor estimates and reactive fluxes characterize rare transitions and reveal mechanisms across proteins and catalytic surfaces. They also show that model expressiveness depends on the chosen basis functions and input features.

Trp-cage folding

  • For Trp-cage, pairwise-distance basis estimates were least sensitive to lag time and avoided boundary artifacts seen in distance-indicator plots.The resulting free energy agreed well with REUS results, and projected reactive fluxes gave pathways and transition states consistent with earlier studies.Source: §1 · p. 6

Insulin dimer dissociation

  • Approximately 100 ms was the estimated insulin-dimer dissociation time from just over 94 µs of data, with four pathways and quantified flux exchange.The estimate was markedly slower than Kramers-based estimates; the authors attribute the difference likely to DGA's greater resolution beyond a two-dimensional barrier projection.Source: §1 · p. 6, Figure 1 · p. 7

Voltage sensing

  • For the voltage-sensing domain, the committor revealed five-state microscopic dynamics and a noncanonical helical-screw mechanism with loosely coupled motions.The displacement-charge free energy suggested a two-state mechanism, while committor minima appeared near q≈0.25, 0.50, and 0.75.Source: §1 · p. 6, Figure 2 · p. 7

Hydrogen dynamics on catalytic surfaces

  • For hydrogen on catalytic surfaces, rates agreed with TST at dilute coverage but deviated at higher coverage as trapping slowed rates and hydrogen interactions made them nonmonotonic.The study considered four surface geometries and coverages from about 0.05 to 1.0 monolayer.Source: §1 · pp. 6, 8

Applications

  • DGA's expressiveness depends on its basis functions and input features, which can be difficult to choose for less-characterized molecules and processes.The review identifies neural networks as an approach that learns representations from data.Source: §1 · p. 8

D. Variational methods for reversible dynamics

For reversible dynamics, variational committor methods replace the generator with finite-lag trajectory averages, but stopping trajectories matters for accurate rates. Fixed-point-residual methods can instead use arbitrary sampling distributions and avoid assuming reversibility or stationary initial conditions.

D. Variational methods for reversible dynamics

  • The finite-τ variational objective approaches the Dirichlet form as τ tends to zero, using a trajectory finite difference in place of the generator.This construction requires reversible dynamics and initial states drawn from the stationary distribution π(x).Source: D. Variational methods for reversible… · p. 8
  • Using unstopped endpoints introduces a systematic error in committor and rate estimates that grows with τ by including segments passing through A or B.Stopping trajectories avoids these unwanted contributions.Source: D. Variational methods for reversible… · p. 8
  • A related finite-data variational update need not be a loss gradient, and omitting full A-to-B crossings shifts the loss minimum away from the transition rate.The omitted contribution leaves the committor unchanged but adds a constant to the loss.Source: D. Variational methods for reversible… · p. 8
  • EVCN time-reverse symmetrization makes its update the exact finite-data gradient, while including A-to-B and B-to-A crossings makes the loss estimate reactive flux.Its minimum is rAB at any τ, yielding committor and rate from one optimization.Source: D. Variational methods for reversible… · p. 8
  • Across Trp-cage, villin, and AIB9, EVCN committors stayed accurate as lag time increased and rates plateaued near empirical values, unlike VCN predictions.At short lag times the two losses performed comparably; VCN committors flattened and rates fell as 1/τ at longer lags.Source: D. Variational methods for reversible… · p. 8, Figure 4 · p. 9

E. Minimizing the fixed-point residual

  • Fixed-point-residual approaches can learn kinetic statistics from trajectory data without an explicit generator or stopped transition operator, reversibility, or stationary-distribution sampling.Arbitrary sampling distributions can concentrate data near transition states.Source: E. Minimizing the fixed-point residual · p. 9

1. The two-trajectory trick

The two-trajectory trick makes a product of conditional trajectory expectations estimable by launching independent trajectories from each initial condition, at the cost of more data. Fixed-point iteration avoids that problematic product and can use one trajectory per initial condition, but its noisy targets and boundary treatment require care.

1. The two-trajectory trick

  • A product of two expectations conditioned on the same initial state cannot generally be estimated consistently from one trajectory per initial condition.The two-trajectory trick uses independent trajectories from each initial condition to estimate the product term in the gradient.Source: §1 · p. 10
  • The two-trajectory trick enables the gradient estimate but requires multiple trajectories per initial condition, unlike most available molecular-dynamics datasets.Generating more than two endpoints and averaging distinct pairs can reduce variance.Source: §1 · p. 10

2. Fixed-point iteration

  • Fixed-point iteration avoids the problematic gradient term by repeatedly fitting the model to targets generated from the stopped transition operator.The target is not available as an explicit function and is sampled for each initial condition.Source: §2 · p. 10
  • With one trajectory per initial condition, noisy unbiased targets require a regression loss whose gradient is linear in the target to give an unbiased gradient estimate.Binary cross-entropy is natural for the committor probability; masking is preferred for boundary conditions because boundary-value errors carry into the converged solution.Source: §2 · p. 10
  • For AIB9, fixed-point iteration with many short trajectories gave committors in excellent agreement with empirical estimates from long trajectories.Remaining variance in target estimates can bias the gradient of the regression objective.Source: §2 · p. 10

F. Subspace acceleration

Fixed-point iteration can converge slowly when the stopped transition operator has a dominant eigenvalue near one. Subspace acceleration learns its leading eigenspace and substantially speeds MFPT convergence for the AIB9 transition.

F. Subspace acceleration

  • When the time to reach A ∪ B is long, the dominant eigenvalue of the stopped transition operator approaches one and fixed-point convergence becomes slow.The operator has no eigenvalue exactly equal to one on functions that vanish on A ∪ B.Source: F. Subspace acceleration · pp. 10–11
  • The accelerated iteration learns the dominant p-dimensional eigenspace, shifting convergence control to the gap between λp−1 and λp.An outer loop rescales the learned functions, for example by QR decomposition, to prevent rank deficiency.Source: F. Subspace acceleration · p. 11
  • For AIB9, subspace acceleration with p = 5 converged to the correct MFPT orders of magnitude faster than basic fixed-point iteration across all tested τ.Figure 5F compares convergence without acceleration (dotted lines) and with acceleration (solid lines).Source: F. Subspace acceleration · p. 11, Figure 5 · p. 11

IV. BEYOND THE MARKOV ASSUMPTION

Reduced-variable descriptions can retain memory even when the underlying molecular dynamics is Markovian. The review contrasts memory-kernel corrections with delay embedding and reports examples where memory corrections substantially reduce the lag time needed for accurate statistics.

IV. BEYOND THE MARKOV ASSUMPTION

  • Projecting onto reduced variables introduces history dependence, and even state-space discretization or finite-basis expansion can introduce memory.A clear separation of timescales can make memory decay quickly, but suitable variables may be difficult to find.Source: §IV · p. 11

A. Mori-Zwanzig formalism

  • Mori–Zwanzig corrections estimate a discrete-time memory kernel recursively from trajectory correlations without assuming its functional form.The formulation neglects the orthogonal-variable term; repeated complementary projections make that remainder smaller when the basis captures most slow dynamics.Source: A. Mori-Zwanzig formalism · p. 12
  • Memory reduced the lag time needed to match reference MFPTs and stationary distributions by about an order of magnitude in a triple-well potential and the AIB9 helix transition.For insulin dimer dissociation, one term beyond the Markov approximation was sufficient.Source: A. Mori-Zwanzig formalism · p. 12

B. Delay embedding

  • Delay embedding augments the state with prior observations until it spans the memory time, making the process approximately Markovian at the cost of more state components.For ξ(Xt) ∈ R^m and n delays, the augmented state has m(n + 1) components.Source: B. Delay embedding · p. 12

C. Augmented TPT

The review connects history-augmented transition path theory with statistical bounds for stopped Markov state models. The bounds identify sampling and locality conditions for practical estimation, while the variance and lag-time analysis clarifies when longer lag times help.

C. Augmented TPT

  • History labels extend transition path theory to sequence-dependent statistics, resolving reactive densities, currents, committors, and conditional MFPTs for reaction steps and pathways.The framework can also address cycles in oscillators and excitable systems, where reactant and product states coincide, provided the augmented process is Markovian.Source: C. Augmented TPT · pp. 12–13

A. Mathematical analysis

  • The stopped-MSM relative-variance bound avoids explicit dependence on condition numbers or expected escape times and is lowered by greater indirect connectivity between states.It is small when initial-state probabilities are not too small, neighboring transitions remain probable, and long jumps are suppressed.Source: A. Mathematical analysis · p. 13

V. CHOOSING THE SAMPLE DISTRIBUTION

  • Avoiding very small sampling probabilities for dynamically important states can reduce the bound; equilibrium-proportional sampling may assign such states exponentially small probability as barriers rise.The review notes that the methods’ requirements differ in whether they permit arbitrary initial-condition distributions or require stationary sampling.Source: A. Mathematical analysis · p. 13, Table I · p. 14

A. Mathematical analysis

  • Under the stated conditions, relative asymptotic variance grows no faster than N^3, hence as the cube of barrier height and of log[1/q(x)] in the analyzed model.One factor of N reflects distributing initial conditions across more states, and two reflect counting state pairs.Source: A. Mathematical analysis · p. 13
  • Stopped Markov state models have polynomial relative-variance cost, whereas shooting’s relative variance grows exponentially with barrier height.Aggregate simulation-time gains are larger still because each shooting trajectory runs until reaching A or B.Source: A. Mathematical analysis · p. 13
  • At large lag times, the bound approaches shooting’s variance, while numerical tests find the smallest relative asymptotic variance at an intermediate lag.Longer lag times also reduce the number of independent trajectory segments available from fixed data, supporting explicit memory correction over simply extending the lag.Source: A. Mathematical analysis · pp. 13–14

B. Sampling strategies

The review describes ways to place initial conditions and trajectory data in transition-relevant regions, including committor-guided adaptation. It also relates committors to optimal control, where a particular drift generates reactive trajectories but can be sensitive to long pathways.

B. Sampling strategies

  • Sampling should cover the transition region rather than follow only the equilibrium distribution, and initial points can be refined adaptively when reaction-relevant regions are inadequately sampled.Existing umbrella-sampling or replica-exchange data can seed sampling; gentle steering is one option when prior data are unavailable.Source: B. Sampling strategies · p. 14
  • Weighted ensemble, forward-flux sampling, transition-interface sampling, milestoning, and nonequilibrium umbrella sampling generate short unbiased trajectory segments for rate estimates.Those segments can also support models of conditional averages satisfying equation (6).Source: B. Sampling strategies · p. 14
  • Committor estimates can guide adaptive sampling, but Dirichlet-form and VCN approaches require stationary-distribution reweighting, whereas other estimators permit concentrating initial points near q ≈ 0.5.The review says the stability and convenience trade-off against sampling flexibility may depend on system properties, including multiple pathways and intermediates.Source: B. Sampling strategies · pp. 14–15

C. Committor and optimal control

  • Doob’s h-transform with the committor as h adds a bias drift that reproduces the unbiased dynamics’ reactive trajectory distribution without reweighting.The correction grows where q is small; an accurate committor can then support estimates of other statistics such as the A-to-B rate.Source: C. Committor and optimal control · p. 15
  • Reactive trajectories under the Doob transform can be long when pathways include intermediate basins, allowing errors to accumulate and shift trajectory distributions and estimated statistics.Among drifts that reliably drive A to B, the Doob drift minimizes the stated variational cost.Source: C. Committor and optimal control · p. 15

VI. OUTLOOK

The review identifies reinforcement learning, geometric graph networks, and generative models as opportunities for kinetic-statistics research. It also highlights open challenges in stability, feature engineering, physical validity, and evaluating methods across diverse systems.

VI. OUTLOOK

  • The review calls for deploying and evaluating kinetic-statistics methods on the diverse chemical systems that motivate their development.It describes progress from estimates for tens of microscopic states to estimates for millions, alongside increased computational power.Source: §VI · p. 15

A. Connecting to reinforcement learning

  • The committor is an undiscounted value function with reward 1 on B and 0 otherwise; DGA corresponds to temporal-difference learning with linear function approximation.The two-trajectory trick is connected to double sampling in reinforcement learning.Source: A. Connecting to reinforcement learning · p. 15
  • Projected fixed-point iterations can lose standard convergence guarantees or diverge in worst cases when sampling distributions mismatch the dynamics, though this does not imply practical instability.The review identifies the stability–statistical-efficiency trade-off as an opportunity for reinforcement-learning tools.Source: A. Connecting to reinforcement learning · pp. 15–16

B. Reducing feature engineering

  • Equivariant geometric graph networks could learn dynamical features directly from atomic coordinates while respecting molecular symmetries, reducing reliance on manually chosen features.Memory and runtime costs constrain their scale, and training stability and interpretability remain open questions.Source: B. Reducing feature engineering · p. 16

C. Generative machine learning

  • Generative models can supply uncorrelated initial points, but their learned distributions may lack physical meaning or omit important states, making reweighting ineffective when overlap is poor.Boltzmann-trained models are unreliable near transition states when those regions are poorly represented in training data.Source: C. Generative machine learning · p. 16

VII. CONCLUSIONS

  • The review concludes that kinetic-statistics methods extend beyond molecular examples, including transitions in atmosphere and ocean data, and calls for systematic comparisons and new approaches.It identifies exchange with broader machine learning as a route to sharpen sampling, model design, and theoretical guarantees.Source: §VII · pp. 16–17
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