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Enhanced Sampling in the Age of Machine Learning: Algorithms and Applications

Kai Zhu, Enrico Trizio, Jintu Zhang, Renling Hu, Linlong Jiang, Tingjun Hou, Luigi Bonati

arXiv:2509.04291v1physics.comp-ph

TL;DR

Molecular dynamics is limited by rare events that occur beyond conventional simulation timescales. This review surveys how machine learning enhances sampling, especially through data-driven collective variables, biasing methods, and applications across molecular domains. It concludes that these advances can improve exploration of complex landscapes and reveal mechanisms underlying rare events.

  • Problem

    Rare events and high-dimensional free-energy landscapes make conventional molecular dynamics insufficient for efficiently sampling important molecular processes.

  • Method

    The review synthesizes machine-learning methods for constructing collective variables, improving biasing schemes, and integrating enhanced sampling across applications.

  • Results

    ML-enhanced sampling facilitates efficient exploration of complex landscapes and uncovers mechanistic insights into degrees of freedom driving rare events across diverse applications.

  • Takeaways & Limitations

    Machine learning is especially influential in constructing low-dimensional yet expressive collective variables for enhanced sampling.

  • Takeaways & Limitations

    Replica exchange may require many intermediate replicas because energy is extensive and neighboring distributions need sufficient overlap.

Abstract

from arXiv · show

Molecular dynamics simulations hold great promise for providing insight into the microscopic behavior of complex molecular systems. However, their effectiveness is often constrained by long timescales associated with rare events. Enhanced sampling methods have been developed to address these challenges, and recent years have seen a growing integration with machine learning techniques. This review provides a comprehensive overview of how they are reshaping the field, with a particular focus on the data-driven construction of collective variables. Furthermore, these techniques have also improved biasing schemes and unlocked novel strategies via reinforcement learning and generative approaches. In addition to methodological advances, we highlight applications spanning different areas such as biomolecular processes, ligand binding, catalytic reactions, and phase transitions. We conclude by outlining future directions aimed at enabling more automated strategies for rare-event sampling.

1 Introduction

Molecular dynamics offers detailed molecular insight but faces two major challenges: accurate yet efficient interaction models and rare events occurring beyond conventional simulation timescales. Machine learning and enhanced sampling address these challenges by improving potential-energy models, identifying collective variables, and accelerating exploration of rare transitions.

  • MD provides time-resolved atomic trajectories that enable direct calculation of thermodynamic and kinetic properties.
  • Ab initio methods accurately describe potential-energy surfaces but are computationally expensive, whereas empirical force fields scale better but may lack fidelity for complex chemical processes.
  • Rare events such as protein folding, drug binding, and phase transitions often occur on timescales far beyond conventional MD.
  • Enhanced sampling accelerates configurational exploration by biasing selected collective variables or increasing the likelihood of rare events.
  • This review examines machine-learning integration with enhanced sampling, including data-driven CV identification, novel biasing schemes, post-processing, and applications across molecular systems.

2 Fundamentals of ML-based enhanced sampling

Atomistic simulations use potential-energy models and statistical-mechanical sampling to study molecular systems, but high-dimensional landscapes and large free-energy barriers make rare transitions difficult to observe. Enhanced sampling methods address this through CV-based biasing, replica-based exploration, or path-space sampling.

  • Atomistic simulations model interactions through a potential-energy surface and use Monte Carlo or MD to sample equilibrium distributions and dynamical information.
  • Collective variables reduce the configuration space and define a lower-dimensional free-energy surface that represents metastable states and transition pathways.
  • Rare transitions are inefficiently sampled because they require crossing large free-energy barriers and may occur on millisecond or longer timescales.
  • CV-based enhanced sampling: CV-based methods introduce a bias potential to explore rarely visited regions while retaining access to unbiased thermodynamics through reweighting.
  • CV-based enhanced sampling: Umbrella sampling combines simulations at fixed bias centers, while adaptive variants update the bias to improve exploration of poorly sampled regions.
  • CV-based enhanced sampling: Metadynamics progressively deposits repulsive Gaussians to fill and flatten the free-energy landscape, whereas ABF estimates free-energy gradients directly.
  • Path sampling: Path sampling generates ensembles of unbiased reactive trajectories in path space rather than modifying the potential-energy surface.

3 Data-driven learning of collective variables

Machine learning addresses the difficulty of designing suitable collective variables by learning low-dimensional representations directly from data. The review organizes approaches by structural information, dimensionality reduction, and physics-based objectives such as dynamical operators and committor functions.

  • Traditional CVs based on physical intuition may miss important modes, while their computational cost becomes unfavorable as the number of CVs increases.
  • ML-based CV methods learn representations from datasets by optimizing model parameters with suitable learning objectives.
  • Learned CVs range from linear combinations of primitive descriptors to geometric graph neural networks operating on atomic coordinates.
  • The review groups methods using structural information, including classification and dimensionality reduction, and methods encoding dynamical operators or committor functions.

3.1 What are good collective variables?

Good collective variables provide compact, physically meaningful representations of reactive processes while respecting system symmetries and supporting smooth biasing. Their quality can also be assessed by how well they capture slow modes and correlate with the committor.

  • Collective variables are functions of atomic coordinates that provide compact representations of reactive processes for analysis and enhanced sampling.
  • CVs should be invariant to global rotations and translations, sometimes permutation of identical atoms, and should be continuous and differentiable for biasing forces.
  • A good CV reduces a 3N-dimensional phase space to ideally one or two dimensions while retaining essential process information.
  • The committor function gives the probability that a configuration reaches a particular metastable state, and good CVs should correlate strongly with it.

3.2 Ingredients of machine learning CVs

Machine-learned collective variables are defined by the system representation, model architecture, dataset, and learning objective. These choices determine how simulation data are encoded, optimized, and used for enhanced sampling.

  • MLCVs combine a system representation, model architecture, dataset, and learning objective into a data-driven CV workflow.The learned CV is trained from MD data and then used to drive enhanced sampling.
  • Input representations must account for physical symmetries, using preprocessing, data augmentation, geometric GNNs, or descriptors.Geometric GNNs encode relational information while maintaining symmetry invariance or equivariance.
  • Descriptors such as Steinhardt parameters, symmetry functions, SOAP, structure-factor peaks, and graph-based features encode local or domain-specific structure.Richer local-environment descriptors provide more information but can require greater computational cost.
  • Architectures range from linear models and kernels to feedforward neural networks and geometric GNNs, trading computational cost against nonlinear expressiveness.
  • Dataset requirements depend on the learning objective, and reliable CV training requires configurations and, when relevant, transitions that represent the system adequately.Unsupervised methods can use unlabeled MD trajectories, whereas supervised methods require labeled data; physics-informed methods often require ergodic or stationary biased simulations.

3.3 Structure-based approaches

Structure-based approaches construct CVs from geometric or topological information using classification, dimensionality reduction, path descriptions, or combinations of objectives. Supervised methods distinguish known metastable states, while unsupervised methods extract representations without explicit targets.

  • Structure-based approaches: Structure-based CV methods use geometric or topological features for state classification, dimensionality reduction, path approximation, or multitask learning.
  • Metastable states classification: LDA separates predefined states by maximizing inter-class variance while minimizing intra-class variance in descriptor space.Its eigenvectors define separating directions, and the number of usable CVs is NS−1 for NS metastable states.
  • Metastable states classification: HLDA replaces LDA covariance calculations with harmonic means to address suboptimal weighting of fluctuations in more stable states.The approach has been applied to biological and chemical systems.
  • Metastable states classification: Deep-LDA applies LDA after a neural network transforms descriptors into a latent space where metastable states can become linearly separable.The network is optimized by maximizing the LDA discrimination score.
  • Metastable states classification: Deep-TDA imposes target distributions on projected training data and can use fewer than NS−1 CVs when additional state-ordering information makes a lower-dimensional representation sufficient.
  • Dimensionality reduction: Unsupervised methods learn continuous differentiable representations without explicit targets, including PCA and autoencoders whose latent spaces can serve as CVs.Autoencoders reconstruct inputs from a low-dimensional bottleneck, providing a nonlinear analogue of PCA for analysis and biasing.

3.4 Physics-based approaches: slow modes

Physics-based CV methods target dynamical information, especially slow modes that govern rare transitions. Forecasting models and dynamical-operator approaches use time-lagged trajectory data to learn representations associated with long-lived processes.

  • Slow modes: Physics-based approaches identify CVs by predicting future configurations, learning dynamical operators, or analyzing transition matrices and spectral structure.
  • Forecasting the dynamics: Time-lagged autoencoders compress configurations at time t into a latent CV space that reconstructs configurations at a future time t+τ.VDE extends this strategy with variational autoencoders and time-lagged reconstruction.
  • Forecasting the dynamics: TAEs and VDEs can encode slow-mode information for enhanced sampling, but they tend to learn mixtures of slow and maximum-variance modes.
  • Dynamical operator learning: Dynamical-operator learning approximates eigenfunctions and eigenvalues of operators such as the Koopman or transfer operator from trajectory time series.Variational formulations construct finite-dimensional approximations within selected trial-function spaces.
  • Dynamical operator learning: VAC identifies trial functions with high time autocorrelation, whose optimized eigenvalue estimates reflect the persistence of dynamical modes.These quantities can be extracted from trajectory data to obtain slow CVs.
  • Dynamical operator learning: TICA, Deep-TICA, VAMPnets, and time-lagged t-SNE extend slow-mode learning through linear, neural-network, variational, or parametric nonlinear representations.Biased trajectories may require time rescaling or reweighting before dynamical analysis.
  • Dynamical operator learning: All dynamical-operator methods require sufficiently informative data because the learned operators encode the system’s long-time dynamics.

3.5 Physics-based approaches: leveraging the committor function

Committor-based methods learn a reaction coordinate tied to the probability of reaching one metastable state before another. They use regression, maximum likelihood, or variational formulations, but informative transition-region data are difficult to obtain.

  • Committor-based approaches: The committor pB(R) gives the probability of reaching state B before A, equaling 0 in A, 1 in B, and interpolating along transition paths.Configurations with pB(R)≈0.5 are associated with the transition-state ensemble.
  • Learning strategies: Committor-learning methods comprise regression on empirical values, maximum likelihood using transition-path data, and variational principles.
  • Regression: Regression methods fit predicted committors to empirical values but require large numbers of computationally expensive committor trajectories.
  • Maximum likelihood: Maximum-likelihood approaches infer committor parameters from transition-path shooting outcomes, often using a sigmoid of a descriptor-based reaction coordinate.Each shooting outcome indicates whether the trajectory reaches B or A.
  • Variational approaches: Variational approaches obtain committor estimates by minimizing functionals based on displacement, reactive flux, or the Kolmogorov backward equation under boundary conditions q=0 in A and q=1 in B.These formulations can be combined with path methods or neural networks.
  • Sampling challenge: Because informative committor behavior is concentrated in the narrow transition region, iterative schemes enhance transition-state sampling to generate useful training data.AIMMD refines shooting-point selection, whereas self-consistent biasing promotes sampling where the committor gradient is large.
  • Sampling challenge: Although theoretically ideal, the committor has nearly constant values in metastable basins and therefore provides vanishing gradients and ineffective biasing forces there.

3.6 Software

MLCV software packages streamline the preprocessing, training, and deployment of machine-learned collective variables in molecular simulations. These tools provide interfaces for diverse model architectures and integration with established MD engines.

  • Software packages address the practical handling of data preprocessing, model training, and MD-engine integration required to implement MLCVs.The review covers tools supporting construction, training, and deployment of MLCVs.
  • mlcolvar provides a unified Python interface for defining, training, and exporting diverse CV models for deployment through PLUMED.Available architectures include feed-forward neural networks, autoencoders, graph neural networks, and multitask models.
  • MLCV integrates neural networks into the C++ Colvars library, while DeepCV implements the DAENN algorithm in Python and C++.MLCV requires exporting neural-network parameters into a text file for native Colvars evaluation.

4 Applications of machine-learned CVs

Machine-learned CVs have been applied to rare conformational changes, ligand binding, phase transformations, and catalytic reactions. Across these systems, they support enhanced sampling, mechanistic interpretation, and dimensionality reduction, but require problem-specific choices of descriptors and models.

  • Biological conformational changes: MLCVs address rare biomolecular transitions by reducing complex free-energy landscapes and revealing mechanistic pathways.Applications include protein folding, transporter conformational changes, DNA translocation, protein dimerization, and mutant stability.
  • Biological conformational changes: A 2D semi-supervised multitask CV combined experimental endpoint states with simulation-identified intermediates to compare two asynchronous Polη DNA-translocation routes.One route moves the primer first; the other moves the template first.
  • Ligand binding: ML-enhanced CVs for ligand binding can estimate free energies and rate constants while supporting mechanistic interpretation of molecular recognition.Studies examined dissociation pathways, metastable intermediates, residence times, and solvent-related slow modes.
  • Structural phase transformations: Effective phase-transition CVs must capture local order and collective structural changes, remain permutationally invariant, and remain robust across intermediates.ML helps process heterogeneous descriptor sets while constructing low-dimensional representations of the transition landscape.
  • Chemical and catalytic reactions: For complex catalytic reactions, supervised CVs can use electronic descriptors to reconstruct free-energy landscapes beyond structural distances and angles.A charge-transfer CV for nitrogen dissociation on iron provided insight into the catalytic role of the surface.
  • MLCV applications pursue varied objectives, so selecting a method requires care rather than relying on a single universal solution.The review identifies enhanced sampling, rare-event exploration, mechanistic insight, and dimensionality reduction as distinct goals.

5 Machine learning bias potentials

Machine learning is being used both to represent high-dimensional free-energy surfaces and to optimize bias potentials or transition-path distributions. These approaches improve scalability, uncertainty-guided exploration, and the flexibility of enhanced sampling beyond fixed low-dimensional biasing.

  • Representing and biasing high-dimensional free energy surfaces: High-dimensional free-energy surfaces are difficult to represent because of the curse of dimensionality and limited simulation data.Kernel methods and neural networks are used to model equilibrium probability distributions and associated free-energy surfaces.
  • Representing and biasing high-dimensional free energy surfaces: Gaussian processes and neural networks provide data-efficient, scalable representations that support free-energy differences, ensemble averages, and bias construction.Gaussian processes incorporate smoothness assumptions and sampling noise, while neural networks can learn free energies or their derivatives.
  • Enhancing biasing schemes with NNs: Reinforced dynamics uses an ensemble uncertainty indicator to bias only regions where the learned free-energy model is sufficiently reliable.The indicator is the standard deviation of mean-force predictions across neural networks.
  • Enhancing biasing schemes with NNs: Adaptive RiD clusters high-uncertainty configurations, labels representatives with restrained MD, and retrains neural-network ensembles using dynamically adjusted thresholds.The adaptive extension was developed because standard RiD performance degraded above 20 CVs.
  • Enhancing biasing schemes with NNs: Neural-network biasing methods such as FUNN and CFF provide smooth force or free-energy estimates, including in sparsely sampled regions.FUNN targets smooth mean-force estimates, while CFF combines force- and frequency-based estimators through a self-integrating neural network.
  • Transition path-guided bias: Transition-path-guided methods learn position- and velocity-dependent bias potentials intended to preserve the statistical properties of unbiased transition-path ensembles.The approaches draw on reinforcement learning, stochastic optimal control, and variational inference rather than predefined CVs.

6 Generative models assist sampling

Generative models assist enhanced sampling by transforming simple distributions into complex molecular targets and by directly generating equilibrium configurations. Their applications include Boltzmann generators, learned free-energy perturbation, and integrations with replica exchange, although complex-system scaling and suitable transformations remain challenging.

  • Generative-model foundations: Normalizing flows learn invertible mappings from simple distributions to complex targets, enabling efficient sampling and exact density estimation.Their tractable change-of-variables formulation requires architectures with easily computed Jacobian determinants, often built from invertible coupling layers.
  • Generative-model foundations: Diffusion models instead generate samples through stochastic forward noising and learned backward denoising, trading exact likelihood evaluation for greater flexibility.The forward process progressively adds noise, while the reverse process reconstructs samples from the original distribution.
  • Boltzmann generators: Boltzmann Generators directly sample equilibrium Boltzmann distributions, bypassing long MD or Monte Carlo simulations and enabling low-probability-state generation.The review notes that this can support continuous free-energy profiles and realistic transition pathways, but explicit solvent, long-range interactions, and flow invertibility constrain complex-system applications.
  • Learned free-energy perturbation: Learned free-energy perturbation represents the transformation M with a normalizing flow trained to minimize expected map work.The approach improves convergence by enhancing overlap and can reuse the full dataset for training and evaluating the free-energy correction, but designing M remains nontrivial.
  • Integrations with replica exchange: Replica exchange connects difficult target distributions with easier ones through information flow across replicas, but extensive energy often requires many intermediate replicas for sufficient overlap.This limitation motivates learned replica exchange, which uses a normalizing flow to transform between prior and target distributions and attempt direct exchanges.
  • Integrations with replica exchange: A denoising diffusion model trained on replica-exchange data learned the joint distribution p(x, T), generating rare low-temperature configurations and extrapolating to temperatures absent from the original ladder.The approach was demonstrated on small peptides and RNA strands as an augmentation of traditional replica exchange.

7 Conclusions

The review finds that machine learning has reshaped enhanced sampling, especially through collective-variable construction, while also advancing biasing, free-energy, replica-exchange, and generative approaches. Applications span diverse molecular problems, but scalability, interpretability, benchmarking, and automation remain important challenges.

  • Collective variables: Machine learning has produced the most substantial advances in enhanced sampling through data-driven construction of collective variables.The review links this progress to ML’s suitability for identifying low-dimensional yet expressive molecular representations.
  • Collective variables: ML-enhanced sampling spans physics-based and structural objectives, with learning choices tightly coupled to the availability and quality of data.Successful workflows often alternate between data collection and iterative CV refinement.
  • Challenges and future directions: The field still lacks a single well-defined objective, and many methods have been tested only on toy or overly simplified systems.The authors argue that rigorous benchmarks and practical baselines are needed for systematic comparison.
  • Biasing and generative approaches: ML contributes beyond CVs by representing bias potentials, optimizing free-energy perturbation, guiding replica exchange, and introducing generative sampling strategies.Approaches that replace conventional biasing or sampling entirely with ML remain at an early stage.
  • Applications: Applications include protein folding, ligand binding, phase transformations, and catalytic reactions, where ML-enhanced sampling supports landscape exploration and mechanistic insight.The review emphasizes insights into the key degrees of freedom associated with rare events.
  • Challenges and future directions: Scaling to intrinsically disordered proteins, biomolecular assemblies, and realistic catalytic environments remains difficult because deployment still requires substantial chemical intuition.Initial conditions, representations, and processes of interest are not yet selected through a fully automated process.
  • Challenges and future directions: Future progress depends on better representations, integrated CV and bias learning, interpretability, closer integration with ML potentials, and modular software ecosystems.The review also calls for rigorous benchmarks and well-defined baselines to distinguish meaningful advances from incremental variants.
  • Outlook: The review envisions ML-enhanced sampling transforming molecular dynamics into a computational microscope across extended time and length scales.This vision concerns atomistic insight into the structure, dynamics, and reactivity of complex physical, chemical, and biological systems.

Biographies

The biographies describe researchers working across pharmaceutics, materials science, physics, chemistry, and molecular modeling, with shared interests in machine learning, molecular dynamics, enhanced sampling, and drug discovery.

  • Researchers: Kai Zhu studies machine learning-based enhanced sampling methods and their applications to atomic systems.
  • Researchers: Enrico Trizio focuses on combining machine learning with molecular dynamics and enhanced sampling simulations.
  • Researchers: Jintu Zhang researches machine learning-based methods for molecular dynamics simulations.
  • Researchers: Renling Hu integrates machine learning, enhanced sampling, quantum mechanics, and free-energy calculations.
  • Researchers: Linlong Jiang studies machine learning-based methods for protein–protein docking and conformational sampling.
  • Researchers: Tingjun Hou works on molecular modeling and AI-driven drug discovery, including virtual screening and pharmacokinetic prediction.
  • Researchers: Luigi Bonati studies the integration of machine learning with enhanced sampling, including potentials for rare events and data-driven CV discovery.
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