We study the fluctuation properties of the local time density, ρ_{T}=1/T∫_{0}^{T}δ(r(t)-1)dt, spent by a d-dimensional Brownian particle at a spherical shell of unit radius, where r(t) denotes the radial distance from the particle to the origin. In the large observation time limit, T→∞, the local time density ρ_{T} obeys the large deviation principle, P(ρ_{T}=ρ)∼e^{-TI(ρ)}, where the rate function I(ρ) is analytic everywhere for d≤4. In contrast, for d>4,I(ρ) becomes nonanalytic at a specific point ρ=ρ_{c}^{(d)}, where ρ_{c}^{(d)}=d(d-4)/(2d-4) depends solely on dimensionality. The singularity signals the occurrence of a first-order dynamical phase transition in dimensions higher than four. Such a transition is accompanied by temporal phase separations in the large deviations of Brownian trajectories. Finally, we validate our theoretical results using a rare-event simulation approach.

Compound multivariate Hawkes processes: Large deviations and rare event simulation

These notes are based on the lectures that one of us (HT) gave at the Summer School on the “Theory of Large Deviations and Applications,” held in July 2024 at Les Houches in France. They present the basic definitions and mathematical results that form the theory of large deviations, as well as many simple motivating examples of applications in statistical physics, which serve as a basis for the many other lectures given at the school that covered more specific applications in biophysics, random matrix theory, nonequilibrium systems, geophysics, and the simulation of rare events, among other topics. These notes extend the lectures, which can be accessed online, by presenting exercises and pointer references for further reading.

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.

Rare events are a common yet challenging topic in many fields of interest. Among many importance-sampling-based rare event simulation methods, forward flux sampling (FFS), established on the effective positive flux framework, is a widely used rare event sampling method due to its simplicity and less restrictive nature. FFS is commonly assumed to work well under diffusive regimes, whereas for correlated systems, the initial flux simulation needs to be either sufficiently long or initiated from multiple uncorrelated starting points to sample a sufficient number of uncorrelated points, and the short timescale recrossing over the energy barrier may result in a potentially overestimated reaction rate. To solve the above-mentioned problems, the author(s) propose an iterative method by regarding the space of the initial outgoing trajectory distribution as the state space of a discrete time Markov Chain (DTMC) process. Upon convergence of the DTMC process by iteratively applying the probability kernel, the initial outgoing trajectory distribution converges to the stationary distribution, and the associated measurable converges to the expected value. Numerical results show that the system is able to converge to its stationary distribution with poor initial outgoing trajectory distribution leading to either orders of magnitude higher or lower estimations of reaction rate compared to its expected value. Meanwhile, the proposed method is able to obtain the recrossing free reaction rate in strongly correlated systems, which could be an order of magnitude smaller than that obtained via the standard FFS method.

Rare event sampling algorithms are essential for understanding processes that occur infrequently on the molecular scale, yet they are important for the long-time dynamics of complex molecular systems. One of these algorithms, transition path sampling (TPS), has become a standard technique to study such rare processes since no prior knowledge on the transition region is required. Most TPS methods generate new trajectories from old trajectories by selecting a point along the old trajectory, modifying its momentum in some way, and then "shooting" a new trajectory by integrating forward and backward in time. In some procedures, the shooting point is selected independently for each trial move, but in others, the shooting point evolves from one path to the next so that successive shooting points are related to each other. To account for this memory effect, we introduce a theoretical framework based on an extended ensemble that includes both paths and shooting indices. We derive appropriate acceptance rules for various path sampling algorithms in this extended formalism, ensuring the correct sampling of the transition path ensemble. Our framework reveals the need for amended acceptance criteria in the flexible-length aimless shooting and spring shooting methods.

ABSTRACTMolecular dynamics simulations are widely used across chemistry, physics, and biology, providing quantitative insight into complex processes with atomic detail. However, their limited timescale of a few microseconds is a significant obstacle in describing phenomena such as conformational transitions of biomolecules and polymorphism in molecular crystals. Recently, stochastic resetting, that is, randomly stopping and restarting the simulations, emerged as a powerful enhanced sampling approach, which is collective variable‐free, highly parallelized, and easily implemented in existing molecular dynamics codes. Resetting expedites sampling rare events while enabling the inference of kinetic observables of the underlying process. It can be employed as a standalone tool or in combination with other enhanced sampling methods, such as Metadynamics, with each technique compensating for the drawbacks of the other. Here, we comprehensively describe resetting and its theoretical background, review recent developments in stochastic resetting for enhanced sampling, and provide instructive guidelines for practitioners.This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte‐Carlo Methods Theoretical and Physical Chemistry > Statistical Mechanics

Abstract. Initialized ensemble simulations can help identify the physical drivers and assess the probabilities of weather and climate extremes based on a given initial state. However, the significant computational burden of complex climate models makes it challenging to quantitatively investigate extreme events with probabilities below a few percent. A possible solution to overcome this problem is to use rare event algorithms, i.e. computational techniques originally developed in statistical physics that increase the sampling efficiency of rare events in numerical simulations. Here, we apply a rare event algorithm to ensemble simulations with the intermediate-complexity coupled climate model PlaSim-LSG to study extremes of pan-Arctic sea ice area reduction under pre-industrial greenhouse gas conditions. We construct four pairs of control and rare event algorithm ensemble simulations, each starting from four different initial winter sea ice states. The rare event simulations produce sea ice lows with probabilities of 2 orders of magnitude smaller than feasible with the control ensembles and drastically increase the number of extremes compared to direct sampling. We find that for a given probability level, the amplitude of negative late-summer sea ice area anomalies strongly depends on the baseline winter sea ice thickness but hardly on the baseline winter sea ice area. Finally, we investigate the physical processes in two trajectories leading to sea ice lows with conditional probabilities of less than 0.001 %. In both cases, negative late-summer pan-Arctic sea ice area anomalies are preceded by negative spring sea ice thickness anomalies. These are related to enhanced surface downward longwave radiative and sensible heat fluxes in an anomalously moist, cloudy and warm atmosphere. During summer, extreme sea ice area reduction is favoured by enhanced open-water-formation efficiency, anomalously strong downward solar radiation and the sea ice–albedo feedback. This work highlights that the most extreme summer sea ice conditions result from the combined effects of preconditioning and weather variability, emphasizing the need for thoughtful ensemble design when turning to real applications.

This article reviews the concepts and methods of variational path sampling. These methods allow computational studies of rare events in systems driven arbitrarily far from equilibrium. Based upon a statistical mechanics of trajectory space and leveraging the theory of large deviations, they provide a perspective from which dynamical phenomena can be studied with the same types of ensemble reweighting ideas that have been used for static equilibrium properties. Applications to chemical, material, and biophysical systems are highlighted.

Importance sampling is a rare event simulation technique used in Monte Carlo simulations to bias the sampling distribution towards the rare event of interest. By assigning appropriate weights to sampled points, importance sampling allows for more efficient estimation of rare events or tails of distributions. However, importance sampling can fail when the proposal distribution does not effectively cover the target distribution. In this work, we propose a method for more efficient sampling by updating the proposal distribution in the latent space of a normalizing flow. Normalizing flows learn an invertible mapping from a target distribution to a simpler latent distribution. The latent space can be more easily explored during the search for a proposal distribution, and samples from the proposal distribution are recovered in the space of the target distribution via the invertible mapping. We empirically validate our methodology on simulated robotics applications such as autonomous racing and aircraft ground collision avoidance.

A major challenge for many rare-event sampling strategies is the identification of progress coordinates that capture the slowest relevant motions. Machine-learning methods that can identify progress coordinates in an unsupervised manner have therefore been of great interest to the simulation community. Here, we developed a general method for identifying progress coordinates "on-the-fly" during weighted ensemble (WE) rare-event sampling via deep learning (DL) of outliers among sampled conformations. Our method identifies outliers in a latent space model of the system's sampled conformations that is periodically trained using a convolutional variational autoencoder. As a proof of principle, we applied our DL-enhanced WE method to simulate the NTL9 protein folding process. To enable rapid tests, our simulations propagated discrete-state synthetic molecular dynamics trajectories using a generative, fine-grained Markov state model. Results revealed that our on-the-fly DL of outliers enhanced the efficiency of WE by >3-fold in estimating the folding rate constant. Our efforts are a significant step forward in the unsupervised learning of slow coordinates during rare event sampling.

BPS2025 - Enhanced sampling of rare events in biomolecular simulations: Conformational transitions and drug unbinding

BPS2025 - Enhanced sampling of rare events in biomolecular simulations: Conformational transitions and drug unbinding

This paper presents a study of rare noise-induced transitions from stable laminar flow to transitional turbulence in plane Couette flow, which we will term buildup. We wish to study forced paths that go all the way from laminar to turbulent flow and to focus the investigation on whether these paths share the properties of noise-induced transitions in simpler systems. The forcing noise has a red spectrum without any component in the natural, large-scale, linear receptivity range of the flow. As we decreased the forcing energy injection rate, the transitions became rare. The rare paths from laminar to turbulent flow are computed using adaptive multilevel splitting, a rare event simulation method, and are validated against direct numerical simulations at moderately small energy injection rates. On the computed trajectories, the flow manages to nonlinearly redistribute energy from the small forced scales to the unforced large scales so that the reactive trajectories display forced streamwise velocity tubes at the natural scale of velocity streaks. As the trajectory proceeds, these tubes gradually grow in amplitude until they cross the separatrix between laminar and turbulent flow. Streamwise vortices manifest themselves only after velocity tubes have reached near-turbulent amplitude, displaying a two-stage process reminiscent of the "backward" path from turbulence to laminar flow. We checked that these were not time-reversed turbulence collapse paths. As the domain size is increased from a minimal flow unit (MFU) type flow at L_{x}×L_{z}=6×4 (in half gap units) to a large domain L_{x}×L_{z}=36×24, spatial localization and then extension of the generated coherent streaks and vortices in the spanwise direction is observed in the reactive paths. The paths systematically computed in MFU display many of the characteristics of instantons that often structure noise-induced transitions: such as concentration of trajectories, exponentially increasing waiting times before transition, and Gumbel distribution of trajectory durations. However, bisections started from successive states on the reactive trajectories indicate that for all sizes and energy injection rates investigated, the trajectory lacks two key ingredients of instantons. First, they do not visit the neighborhood of the nearest saddle point and do not display the natural relaxation path from that saddle to transitional wall turbulence. This discrepancy is observed for all system sizes. Second, the reactive paths do not concentrate more and more around the same trajectory as energy injection rate is decreased, but instead gradually move in phase space. They might reconnect with instantons at very small energy injection rate and exceedingly long waiting times. They would explain why classical instanton calculations have proved to be tremendously difficult in wall flows.

Studying the kinetics of long-timescale rare events is a fundamental challenge in molecular simulation. To address this problem, we propose an integration of two different rare-event sampling philosophies: biased enhanced sampling and unbiased path sampling. Enhanced sampling methods, e.g., metadynamics, can facilitate the crossing of free energy barriers by applying an external bias potential. On the contrary, path sampling methods like weighted ensemble do not apply any biasing force but accelerate the exploration of rugged free energy surfaces through trajectory resampling. We show that a judicious combination of the weighted ensemble with a metadynamics-like algorithm can synergize the strengths and mitigate the deficiencies of path sampling and enhanced sampling approaches. The resulting integrated sampling (IS) algorithm improves the computational efficiency of calculating the kinetics of peptide conformational transitions, protein unfolding, and the dissociation of a ligand-receptor complex. Furthermore, the IS approach can direct sampling along the minimum free energy pathway even when the collective variable used for biasing is suboptimal. These advantages make the IS algorithm suitable for studying the kinetics of complex molecular systems of biological and pharmaceutical relevance.

Locating plausible transition paths and enhanced sampling of rare events are fundamental to understanding the functional dynamics of biomolecules. Here, a constraint-based constant advance replicas (CAR) formalism of reaction paths is reported for identifying the most probable transition path (MPTP) between two given states. We derive the temporal-integrated effective dynamics governing the projected subsystem under the holonomic CAR path constraints and show that a dynamical action functional can be defined and used for optimizing the MPTP. We further demonstrate how the CAR MPTP can be located by asymptotically minimizing an upper bound of the CAR action functional using a variational expectation-maximization framework. Essential thermodynamics and kinetic observables are retrieved by integrating the boxed molecular dynamics on the CAR MPTP using a newly proposed adaptive reflecting boundary condition. The efficiency of the proposed method is demonstrated for the Müller potential, the alanine dipeptide isomerization, and the DNA base pairing transition (Watson-Crick to Hoogsteen) in explicit solvent. The CAR representation of transition paths constitutes a robust and extensible platform that can be combined with diverse enhanced sampling methods to aid future flexible and reliable biomolecular simulations.

From physics and biology to seismology and economics, the behaviour of countless systems is determined by impactful yet unlikely transitions between metastable states known as rare events, the study of which is essential for understanding and controlling the properties of these systems. Classical computational methods to sample rare events remain prohibitively inefficient and are bottlenecks for enhanced samplers that require prior data. Here we introduce a physics-informed machine learning framework, normalizing Flow enhanced Rare Event Sampler (FlowRES), which uses unsupervised normalizing flow neural networks to enhance Monte Carlo sampling of rare events by generating high-quality non-local Monte Carlo proposals. We validated FlowRES by sampling the transition path ensembles of equilibrium and non-equilibrium systems of Brownian particles, exploring increasingly complex potentials. Beyond eliminating the requirements for prior data, FlowRES features key advantages over established samplers: no collective variables need to be defined, efficiency remains constant even as events become increasingly rare and systems with multiple routes between states can be straightforwardly simulated.

We present a flow-based method for simulating and calculating nucleation rates of first-order phase transitions in scalar field theory on a lattice. Motivated by recent advancements in machine learning tools, particularly normalizing flows for lattice field theory, we propose the “partitioning flow-based Markov chain Monte Carlo (PFMCMC) sampling” method to address two challenges encountered in normalizing flow applications for lattice field theory: the “mode-collapse” and “rare-event sampling” problems. Using a (2+1)-dimensional real scalar model as an example, we demonstrate the effectiveness of our PFMCMC method in modeling highly hierarchical order parameter probability distributions and simulating critical bubble configurations. These simulations are then used to facilitate the calculation of nucleation rates. We anticipate the application of this method to (3+1)-dimensional theories for studying realistic cosmological phase transitions.

The folding and unfolding of RNA stem-loops are critical biological processes; however, their computational studies are often hampered by the ruggedness of their folding landscape, necessitating long simulation times at the atomistic scale. Here, we adapted DeepDriveMD (DDMD), an advanced deep learning-driven sampling technique originally developed for protein folding, to address the challenges of RNA stem-loop folding. Although tempering- and order parameter-based techniques are commonly used for similar rare-event problems, the computational costs or the need for a priori knowledge about the system often present a challenge in their effective use. DDMD overcomes these challenges by adaptively learning from an ensemble of running MD simulations using generic contact maps as the raw input. DeepDriveMD enables on-the-fly learning of a low-dimensional latent representation and guides the simulation toward the undersampled regions while optimizing the resources to explore the relevant parts of the phase space. We showed that DDMD estimates the free energy landscape of the RNA stem-loop reasonably well at room temperature. Our simulation framework runs at a constant temperature without external biasing potential, hence preserving the information on transition rates, with a computational cost much lower than that of the simulations performed with external biasing potentials. We also introduced a reweighting strategy for obtaining unbiased free energy surfaces and presented a qualitative analysis of the latent space. This analysis showed that the latent space captures the relevant slow degrees of freedom for the RNA folding problem of interest. Finally, throughout the manuscript, we outlined how different parameters are selected and optimized to adapt DDMD for this system. We believe this compendium of decision-making processes will help new users adapt this technique for the rare-event sampling problems of their interest.

A dynamic importance function for accidental scenarios generation by RESTART in the computational risk assessment of cyber-physical infrastructures