The sampling of Boltzmann distributions by stochastic Markov processes, can be strongly limited by the crossing time of high (free) energy barriers. As a result, the system may stay trapped in metastable states, and the relaxation time to the equilibrium Boltzmann distribution may be very large compared to the available computational time. In this paper, we show how, by a simple modification of the Hamiltonian, one can dramatically decrease the relaxation time of the system, while retaining the same equilibrium distribution. The method is illustrated on the case of the one-dimensional double-well potential.
Laboratory of Physics of the \'Ecole Normale Sup\'erieure, CNRS UMR 8023 \& PSL Research, Sorbonne Universit\'e, 24 rue Lhomond, 75005 Paris, France
Restricted Boltzmann Machines (RBM) are bi-layer neural networks used for the unsupervised learning of model distributions from data. The bipartite architecture of RBM naturally defines an elegant sampling procedure, called Alternating Gibbs Sampling (AGS), where the configurations of the latent-variable layer are sampled conditional to the data-variable layer, and vice versa. We study here the performance of AGS on several analytically tractable models borrowed from statistical mechanics. We show that standard AGS is not more efficient than classical Metropolis-Hastings (MH) sampling of the e…
Boltzmann Sampling of Frustrated J1 - J2 Ising Models with Programmable Quantum Annealers
Information Systems & Modeling, Los Alamos National Laboratory（信息系统与建模，洛斯阿拉莫斯国家实验室）；Theoretical Division, Quantum & Condensed Matter Physics, Los Alamos National Laboratory（理论部，量子与凝聚态物理，洛斯阿拉莫斯国家实验室）；Center for Quantum Computing, Los Alamos National Laboratory（量子计算中心，洛斯阿拉莫斯国家实验室）
Boltzmann sampling with quantum annealers via fast Stein correction
Graduate School of Frontier Sciences, The University of Tokyo（东京大学前沿科学研究生院）；Center for Basic Research on Materials, National Institute for Materials Science（物质材料研究机构基础材料研究中心）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
FU Berlin（柏林自由大学）；Rice University（莱斯大学）；Tongji University（同济大学）
Computing equilibrium states in condensed-matter many-body systems, such as solvated proteins, is a long-standing challenge. Lacking methods for generating statistically independent equilibrium samples in "one shot", vast computational effort is invested for simulating these system in small steps, e.g., using Molecular Dynamics. Combining deep learning and statistical mechanics, we here develop Boltzmann Generators, that are shown to generate unbiased one-shot equilibrium samples of representative condensed matter systems and proteins. Boltzmann Generators use neural networks to learn a coordi…
Universal representation by Boltzmann machines with Regularised Axons
ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology（ICFO——光子科学研究所，巴塞罗那科学技术研究院）；Adam Mickiewicz University（亚当·密茨凯维奇大学）；Quside Technologies SL（Quside科技有限公司）；Universitat Politècnica de València（瓦伦西亚理工大学）；ICREA - Institució Catalana de Recerca i Estudis Avançats（ICREA——加泰罗尼亚研究与高级研究机构）；University of Innsbruck（因斯布鲁克大学）
Boltzmann generators approach the sampling problem in many-body physics by combining a normalizing flow and a statistical reweighting method to generate samples in thermodynamic equilibrium. The equilibrium distribution is usually defined by an energy function and a thermodynamic state. Here we propose temperature-steerable flows (TSF) which are able to generate a family of probability densities parametrized by a choosable temperature parameter. TSFs can be embedded in generalized ensemble sampling frameworks to sample a physical system across multiple thermodynamic states.
Boltzmann sampling based on Metropolis algorithm has been extensively used for simulating a canonical ensemble. An estimate of a mechanical property, like energy, of an equilibrium system, can be made by averaging over a large number microstates generated by Boltzmann Monte Carlo methods. However, a thermal property like entropy is not easily accessible to these methods. The reason is simple. We can assign a numerical value for energy to each microstate. But we can not assign a numerical value for entropy, to a microstate. Entropy is not a property associated with any single microstate.It is a…
Universidad Complutense de Madrid（马德里康普顿斯大学）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；INRIA（法国国家信息与自动化研究所）
Restricted Boltzmann Machines are simple and powerful generative models that can encode any complex dataset. Despite all their advantages, in practice the trainings are often unstable and it is difficult to assess their quality because the dynamics are affected by extremely slow time dependencies. This situation becomes critical when dealing with low-dimensional clustered datasets, where the time required to sample ergodically the trained models becomes computationally prohibitive. In this work, we show that this divergence of Monte Carlo mixing times is related to a phenomenon of phase coexis…
Scalable Boltzmann Generators for equilibrium sampling of large-scale materials
Microsoft Research AI for Science（微软研究院人工智能科学部门）；Department of Chemistry, Rice University（理查德·伊尔大学化学系）；Initiative for Computational Catalysis, Flatiron Institute（计算催化倡议，Flatiron研究所）
Professorship of Multiscale Modeling of Fluid Materials, Department of Engineering Physics and Computation, TUM School of Engineering and Design, Technical University of Munich, Germany（多尺度流体材料建模教授职位，工程物理与计算系，TUM工程与设计学院，慕尼黑技术大学，德国）；Atomistic Modeling Center, Munich Data Science Institute, Technical University of Munich, Germany（原子建模中心，慕尼黑数据科学研究所，慕尼黑技术大学，德国）
Fast training and sampling of Restricted Boltzmann Machines
INRIA-Saclay（INRIA-萨克莱实验室）；LISN Paris-Saclay University（巴黎萨克雷大学LISN实验室）；Escuela Técnica Superior de Ingenieros Industriales Universidad Politécnica de Madrid（马德里理工大学工业工程师学院）；Departamento de Física Teórica Universidad Complutense de Madrid（马德里complutense大学理论物理系）；LCQB, Sorbonne Université, Laboratoire de Physique Théorique École Normale Supérieure, Paris（巴黎高等师范学院理论物理实验室，索邦大学LCQB）
Temperature Accelerated Sliced Sampling (TASS) is an efficient method to compute high dimensional free energy landscapes. The original TASS method employs the Weighted Histogram Analysis Method (WHAM) which is an iterative post-processing to reweight and stitch high dimensional probability distributions in sliced windows that are obtained in the presence of restraining biases. The WHAM necessitates that TASS windows lie close to each other for proper overlap of distributions and span the collective variable space of interest. On the other hand, increase in number of TASS windows implies more n…
Consider a one-parameter family of Boltzmann distributions $p_t(x) = \tfrac{1}{Z_t}e^{-S_t(x)}$. This work studies the problem of sampling from $p_{t_0}$ by first sampling from $p_{t_1}$ and then applying a transformation $Ψ_{t_1}^{t_0}$ so that the transformed samples follow $p_{t_0}$. We derive an equation relating $Ψ$ and the corresponding family of unnormalized log-likelihoods $S_t$. The utility of this idea is demonstrated on the $ϕ^4$ lattice field theory by extending its defining action $S_0$ to a family of actions $S_t$ and finding a $τ$ such that normalizing flows perform better at le…
Boltzmann generators for amorphous particle systems
Laboratoire de Physique de l’École normale supérieure ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris（巴黎高等师范学院物理实验室，巴黎科学研究院，法国国家科学研究中心，索邦大学，巴黎大学）；CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris（巴黎高等理工学院计算数学与应用物理实验室，法国国家科学研究中心，巴黎理工学院，巴黎理工研究院）；Dipartimento di Fisica, Università di Trieste（特伦蒂诺-上阿迪杰大学物理系）；Gulliver, CNRS UMR 7083, ESPCI Paris, PSL Research University（Gulliver，法国国家科学研究中心UMR 7083，巴黎高等师范学院，巴黎科学研究院）
