arXiv subjects

Gabriel Stoltz

Publications and source records attributed to Gabriel Stoltz.

At least 37 records · Page 2Linked to original sources

arXiv2024

Sampling metastable systems using collective variables and Jarzynski-Crooks paths

We consider the problem of sampling a high dimensional multimodal target probability measure. We assume that a good proposal kernel to move only a subset of the degrees of freedoms (also known as collective variables) is known a priori. This proposal kernel can for example be built using normalizing flows. We show how to extend the move from the collective variable space to the full space and how to implement an accept-reject step in order to get a reversible chain with respect to a target probability measure. The accept-reject step does not require to know the marginal of the original measure in the collective variable (namely to know the free energy). The obtained algorithm admits several variants, some of them being very close to methods which have been proposed previously in the literature. We show how the obtained acceptance ratio can be expressed in terms of the work which appears in the Jarzynski-Crooks equality, at least for some variants. Numerical illustrations demonstrate the efficiency of the approach on various simple test cases, and allow us to compare the variants of the algorithm.

cond-mat.stat-mech↗

arXiv2024

Sticky coupling as a control variate for sensitivity analysis

We present and analyze a control variate strategy based on couplings to reduce the variance of finite difference estimators of sensitivity coefficients, called transport coefficients in the physics literature. We study the bias and variance of a sticky-coupling and a synchronous-coupling based estimator as the finite difference parameter $η$ goes to zero. For diffusions with elliptic additive noise, we show that when the drift is contractive outside a compact the bias of a sticky-coupling based estimator is bounded as $η\to 0$ and its variance behaves like $η^{-1}$, compared to the standard estimator whose bias and variance behave like $η^{-1}$ and $η^{-2}$, respectively. Under the stronger assumption that the drift is contractive everywhere, we additionally show that the bias and variance of the synchronous-coupling based estimator are both bounded as $η\to 0$. Our hypotheses include overdamped Langevin dynamics with many physically relevant non-convex potentials. We illustrate our theoretical results with numerical examples, including overdamped Langevin dynamics with a highly non-convex Lennard-Jones potential to demonstrate both failure of synchronous coupling and the effectiveness of sticky coupling in the not globally contractive setting.

math.PR↗

arXiv2024

Second order quantitative bounds for unadjusted generalized Hamiltonian Monte Carlo

This paper provides a convergence analysis for generalized Hamiltonian Monte Carlo samplers, a family of Markov Chain Monte Carlo methods based on leapfrog integration of Hamiltonian dynamics and kinetic Langevin diffusion, that encompasses the unadjusted Hamiltonian Monte Carlo method. Assuming that the target distribution $π$ satisfies a log-Sobolev inequality and mild conditions on the corresponding potential function, we establish quantitative bounds on the relative entropy of the iterates defined by the algorithm, with respect to $π$. Our approach is based on a perturbative and discrete version of the modified entropy method developed to establish hypocoercivity for the continuous-time kinetic Langevin process. As a corollary of our main result, we are able to derive complexity bounds for the class of algorithms at hand. In particular, we show that the total number of iterations to achieve a target accuracy $\varepsilon >0$ is of order $d/\varepsilon^{1/4}$, where $d$ is the dimension of the problem. This result can be further improved in the case of weakly interacting mean field potentials, for which we find a total number of iterations of order $(d/\varepsilon)^{1/4}$.

math.PR↗

arXiv2024

A spectral approach to the narrow escape problem in the disk

We study the narrow escape problem in the disk, which consists in identifying the first exit time and first exit point distribution of a Brownian particle from the ball in dimension 2, with reflecting boundary conditions except on small disjoint windows through which it can escape. This problem is motivated by practical questions arising in various scientific fields (in particular cellular biology and molecular dynamics). We apply the quasi-stationary distribution approach to metastability, which requires to study the eigenvalue problem for the Laplacian operator with Dirichlet boundary conditions on the small absorbing part of the boundary, and Neumann boundary conditions on the remaining reflecting part. We obtain rigorous asymptotic estimates of the first eigenvalue and of the normal derivative of the associated eigenfunction in the limit of infinitely small exit regions, which yield asymptotic estimates of the first exit time and first exit point distribution starting from the quasi-stationary distribution within the disk.

math.AP↗

arXiv2024

Analyzing multimodal probability measures with autoencoders

Finding collective variables to describe some important coarse-grained information on physical systems, in particular metastable states, remains a key issue in molecular dynamics. Recently, machine learning techniques have been intensively used to complement and possibly bypass expert knowledge in order to construct collective variables. Our focus here is on neural network approaches based on autoencoders. We study some relevant mathematical properties of the loss function considered for training autoencoders, and provide physical interpretations based on conditional variances and minimum energy paths. We also consider various extensions in order to better describe physical systems, by incorporating more information on transition states at saddle points, and/or allowing for multiple decoders in order to describe several transition paths. Our results are illustrated on toy two dimensional systems and on alanine dipeptide.

physics.chem-ph↗

arXiv2023

Hypocoercivity properties of adaptive Langevin dynamics

Adaptive Langevin dynamics is a method for sampling the Boltzmann-Gibbs distribution at prescribed temperature in cases where the potential gradient is subject to stochastic perturbation of unknown magnitude. The method replaces the friction in underdamped Langevin dynamics with a dynamical variable, updated according to a negative feedback loop control law as in the Nosé-Hoover thermostat. Using a hypocoercivity analysis we show that the law of Adaptive Langevin dynamics converges exponentially rapidly to the stationary distribution, with a rate that can be quantified in terms of the key parameters of the dynamics. This allows us in particular to obtain a central limit theorem with respect to the time averages computed along a stochastic path. Our theoretical findings are illustrated by numerical simulations involving classification of the MNIST data set of handwritten digits using Bayesian logistic regression.

math.PR↗

arXiv2023

Friction-adaptive descent: a family of dynamics-based optimization methods

We describe a family of descent algorithms which generalizes common existing schemes used in applications such as neural network training and more broadly for optimization of smooth functions--potentially for global optimization, or as a local optimization method to be deployed within global optimization schemes like basin hopping. By introducing an auxiliary degree of freedom we create a dynamical system with improved stability, reducing oscillatory modes and accelerating convergence to minima. The resulting algorithms are simple to implement and control, and convergence can be shown directly by Lyapunov's second method. Although this framework, which we refer to as friction-adaptive descent (FAD), is fairly general, we focus most of our attention here on a specific variant: kinetic energy stabilization (which can be viewed as a zero-temperature Nosé--Hoover scheme but with added dissipation in both physical and auxiliary variables), termed KFAD (kinetic FAD). To illustrate the flexibility of the FAD framework we consider several other methods. in certain asymptotic limits, these methods can be viewed as introducing cubic damping in various forms; they can be more efficient than linearly dissipated Hamiltonian dynamics in common optimization settings. We present details of the numerical methods and show convergence for both the continuous and discretized dynamics in the convex setting by constructing Lyapunov functions. The methods are tested using a toy model (the Rosenbrock function). We also demonstrate the methods for structural optimization for atomic clusters in Lennard--Jones and Morse potentials. The experiments show the relative efficiency and robustness of FAD in comparison to linearly dissipated Hamiltonian dynamics.

math.OC↗

arXiv2023

Unbiasing Hamiltonian Monte Carlo algorithms for a general Hamiltonian function

Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo method that allows to sample high dimensional probability measures. It relies on the integration of the Hamiltonian dynamics to propose a move which is then accepted or rejected thanks to a Metropolis procedure. Unbiased sampling is guaranteed by the preservation by the numerical integrators of two key properties of the Hamiltonian dynamics: volume-preservation and reversibility up to momentum reversal. For separable Hamiltonian functions, some standard explicit numerical schemes, such as the Störmer-Verlet integrator, satisfy these properties. However, for numerical or physical reasons, one may consider a Hamiltonian function which is nonseparable, in which case the standard numerical schemes which preserve the volume and satisfy reversibility up to momentum reversal are implicit. When implemented in practice, such implicit schemes may admit many solutions or none, especially when the timestep is too large. We show here how to enforce the numerical reversibility, and thus unbiasedness, of HMC schemes in this context by introducing a reversibility check. In addition, for some specific forms of the Hamiltonian function, we discuss the consistency of these HMC schemes with some Langevin dynamics, and show in particular that our algorithm yields an efficient discretization of the metropolized overdamped Langevin dynamics with position-dependent diffusion coefficients. Numerical results illustrate the relevance of the reversibility check on simple problems.

math.NA↗

arXiv2023

Autoencoders for dimensionality reduction in molecular dynamics: collective variable dimension, biasing and transition states

The heat shock protein 90 (Hsp90) is a molecular chaperone that controls the folding and activation of client proteins using the free energy of ATP hydrolysis. The Hsp90 active site is in its N-terminal domain (NTD). Our goal is to characterize the dynamics of NTD using an autoencoder-learned collective variable (CV) in conjunction with adaptive biasing force (ABF) Langevin dynamics. Using dihedral analysis, we cluster all available experimental Hsp90 NTD structures into distinct native states. We then perform unbiased molecular dynamics (MD) simulations to construct a dataset that represents each state and use this dataset to train an autoencoder. Two autoencoder architectures are considered, with one and two hidden layers respectively, and bottlenecks of dimension $k$ ranging from 1 to 10. We demonstrate that the addition of an extra hidden layer does not significantly improve the performance, while it leads to complicated CVs that increases the computational cost of biased MD calculations. In addition, a 2D bottleneck can provide enough information of the different states, while the optimal bottleneck dimension is five. For the 2D bottleneck, the two-dimensional CV is directly used in biased MD simulations. For the 5D bottleneck, we perform an analysis of the latent CV space and identify the pair of CV coordinates that best separates the states of Hsp90. Interestingly, selecting a 2D CV out of the 5D CV space leads to better results than directly learning a 2D CV, and allows to observe transitions between native states when running free energy biased dynamics.

physics.comp-ph↗

arXiv2023

Fixing the flux: A dual approach to computing transport coefficients

We present a method to compute transport coefficients in molecular dynamics. Transport coefficients quantify the linear dependencies of fluxes in non-equilibrium systems subject to small external forcings. Whereas standard non-equilibrium approaches fix the forcing and measure the average flux induced in the system driven out of equilibrium, a dual philosophy consists in fixing the value of the flux, and measuring the average magnitude of the forcing needed to induce it. A deterministic version of this approach, named Norton dynamics, was studied in the 1980s by Evans and Morris. In this work, we introduce a stochastic version of this method, first developing a general formal theory for a broad class of diffusion processes, and then specializing it to underdamped Langevin dynamics, which are commonly used for molecular dynamics simulations. We provide numerical evidence that the stochastic Norton method provides an equivalent measure of the linear response, and in fact demonstrate that this equivalence extends well beyond the linear response regime. This work raises many intriguing questions, both from the theoretical and the numerical perspectives.

cond-mat.stat-mech↗

arXiv2023

Uniform minorization condition and convergence bounds for discretizations of kinetic Langevin dynamics

We study the convergence in total variation and $V$-norm of discretization schemes of the underdamped Langevin dynamics. Such algorithms are very popular and commonly used in molecular dynamics and computational statistics to approximatively sample from a target distribution of interest. We show first that, for a very large class of schemes, a minorization condition uniform in the stepsize holds. This class encompasses popular methods such as the Euler-Maruyama scheme and the schemes based on splitting strategies. Second, we provide mild conditions ensuring that the class of schemes that we consider satisfies a geometric Foster--Lyapunov drift condition, again uniform in the stepsize. This allows us to derive geometric convergence bounds, with a convergence rate scaling linearly with the stepsize. This kind of result is of prime interest to obtain estimates on norms of solutions to Poisson equations associated with a given numerical method.

math.PR↗

arXiv2023

Computing Surface Reaction Rates by Adaptive Multilevel Splitting Combined with Machine Learning and Ab Initio Molecular Dynamics

Computing accurate rate constants for catalytic events occurring at the surface of a given material represents a challenging task with multiple potential applications in chemistry. To address this question, we propose an approach based on a combination of the rare event sampling method called Adaptive Multilevel Splitting (AMS) and ab initio molecular dynamics (AIMD). The AMS method requires a one dimensional reaction coordinate to index the progress of the transition. Identifying a good reaction coordinate is difficult, especially for high dimensional problems such a those encountered in catalysis. We probe various approaches to build reaction coordinates such as Support Vector Machine and path collective variables. The AMS is implemented so as to communicate with a DFT-plane wave code. A relevant case study in catalysis: the change of conformation and the dissociation of a water molecule chemisorbed on the (100) $γ$-alumina surface is used to evaluate our approach. The calculated rate constants and transition mechanisms are discussed and compared to those obtained by a conventional static approach based on the Eyring-Polanyi equation with harmonic approximation. It is revealed that the AMS method may provide rate constants which are smaller than the static approach by up to two orders of magnitude due to entropic effects involved in the chemisorbed water.

physics.chem-ph↗

arXiv2023

Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations

Using Brownian dynamics simulations, we investigate the effects of confinement, adsorption on surfaces and ion-ion interactions on the response of confined electrolyte solutions to oscillating electric fields in the direction perpendicular to the confining walls. Nonequilibrium simulations allow to characterize the transitions between linear and nonlinear regimes when varying the magnitude and frequency of the applied field but the linear response, characterized by the frequency-dependent conductivity, is more efficiently predicted from the equilibrium current fluctuations. To that end, we (rederive and) use the Green-Kubo relation appropriate for overdamped dynamics, which differs from the standard one for Newtonian or underdamped Langevin dynamics. This expression highlights the contributions of the underlying Brownian fluctuations and of the interactions of the particles between them and with external potentials. While already known in the literature, this relation has rarely been used to date, beyond the static limit to determine the effective diffusion coefficient or the DC conductivity. The frequency-dependent conductivity always decays from a bulk-like behavior at high frequency to a vanishing conductivity at low frequency due to the confinement of the charge carriers by the walls. We discuss the characteristic features of the crossover between the two regimes, most importantly how the crossover frequency depends on the confining distance and the salt concentration, and the fact that adsorption on the walls may lead to significant changes both at high- and low-frequencies. Conversely, our results illustrate the possibility to obtain information on diffusion between walls, charge relaxation and adsorption by analyzing the frequency-dependent conductivity.

physics.chem-ph↗

arXiv2023

Extending the regime of linear response with synthetic forcings

Transport coefficients, such as the mobility, thermal conductivity and shear viscosity, are quantities of prime interest in statistical physics. At the macroscopic level, transport coefficients relate an external forcing of magnitude $η$, with $η\ll 1$, acting on the system to an average response expressed through some steady-state flux. In practice, steady-state averages involved in the linear response are computed as time averages over a realization of some stochastic differential equation. Variance reduction techniques are of paramount interest in this context, as the linear response is scaled by a factor of $1/η$, leading to large statistical error. One way to limit the increase in the variance is to allow for larger values of $η$ by increasing the range of values of the forcing for which the nonlinear part of the response is sufficiently small. In theory, one can add an extra forcing to the physical perturbation of the system, called synthetic forcing, as long as this extra forcing preserves the invariant measure of the reference system. The aim is to find synthetic perturbations allowing to reduce the nonlinear part of the response as much as possible. We present a mathematical framework for quantifying the quality of synthetic forcings, in the context of linear response theory, and discuss various possible choices for them. Our findings are illustrated with numerical results in low-dimensional systems.

math.NA↗

arXiv2023

Quantifying the mini-batching error in Bayesian inference for Adaptive Langevin dynamics

Bayesian inference allows to obtain useful information on the parameters of models, either in computational statistics or more recently in the context of Bayesian Neural Networks. The computational cost of usual Monte Carlo methods for sampling posterior laws in Bayesian inference scales linearly with the number of data points. One option to reduce it to a fraction of this cost is to resort to mini-batching in conjunction with unadjusted discretizations of Langevin dynamics, in which case only a random fraction of the data is used to estimate the gradient. However, this leads to an additional noise in the dynamics and hence a bias on the invariant measure which is sampled by the Markov chain. We advocate using the so-called Adaptive Langevin dynamics, which is a modification of standard inertial Langevin dynamics with a dynamical friction which automatically corrects for the increased noise arising from mini-batching. We investigate the practical relevance of the assumptions underpinning Adaptive Langevin (constant covariance for the estimation of the gradient, Gaussian minibatching noise), which are not satisfied in typical models of Bayesian inference, and quantify the bias induced by minibatching in this case. We also suggest a possible extension of AdL to further reduce the bias on the posterior distribution, by considering a dynamical friction depending on the current value of the parameter to sample.

stat.ML↗

arXiv2023

Generative methods for sampling transition paths in molecular dynamics

Molecular systems often remain trapped for long times around some local minimum of the potential energy function, before switching to another one -- a behavior known as metastability. Simulating transition paths linking one metastable state to another one is difficult by direct numerical methods. In view of the promises of machine learning techniques, we explore in this work two approaches to more efficiently generate transition paths: sampling methods based on generative models such as variational autoencoders, and importance sampling methods based on reinforcement learning.

stat.ML↗

arXiv2022

Error estimates and variance reduction for nonequilibrium stochastic dynamics

Equilibrium properties in statistical physics are obtained by computing averages with respect to Boltzmann-Gibbs measures, sampled in practice using ergodic dynamics such as the Langevin dynamics. Some quantities however cannot be computed by simply sampling the Boltzmann-Gibbs measure, in particular transport coefficients, which relate the current of some physical quantity of interest to the forcing needed to induce it. For instance, a temperature difference induces an energy current, the proportionality factor between these two quantities being the thermal conductivity. From an abstract point of view, transport coefficients can also be considered as some form of sensitivity analysis with respect to an added forcing to the baseline dynamics. There are various numerical techniques to estimate transport coefficients, which all suffer from large errors, in particular large statistical errors. This contribution reviews the most popular methods, namely the Green-Kubo approach where the transport coefficient is expressed as some time-integrated correlation function, and the approach based on longtime averages of the stochastic dynamics perturbed by an external driving (so-called nonequilibrium molecular dynamics). In each case, the various sources of errors are made precise, in particular the bias related to the time discretization of the underlying continuous dynamics, and the variance of the associated Monte Carlo estimators. Some recent alternative techniques to estimate transport coefficients are also discussed.

math.NA↗

arXiv2022

Ab initio Canonical Sampling based on Variational Inference

Finite temperature calculations, based on ab initio molecular dynamics (AIMD) simulations, are a powerful tool able to predict material properties that cannot be deduced from ground state calculations. However, the high computational cost of AIMD limits its applicability for large or complex systems. To circumvent this limitation we introduce a new method named Machine Learning Assisted Canonical Sampling (MLACS), which accelerates the sampling of the Born--Oppenheimer potential surface in the canonical ensemble. Based on a self-consistent variational procedure, the method iteratively trains a Machine Learning Interatomic Potential to generate configurations that approximate the canonical distribution of positions associated with the ab initio potential energy. By proving the reliability of the method on anharmonic systems, we show that the method is able to reproduce the results of AIMD with an ab initio accuracy at a fraction of its computational cost.

cond-mat.mtrl-sci↗