Statistical Learning for High-Dimensional Stochastic Dynamical Systems

高维随机动力系统的统计学习

• 批准号: 1522651
• 负责人: Mauro Maggioni
• 金额: $ 30万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522651&HistoricalAwards=false
• 关键词: Statistical; Learning; Dimensional; Stochastic; Dynamical

High-dimensional stochastic dynamical systems arise in a wide variety of scientific fields and applications, including models for the dynamics of molecules, of multi-agent systems (such as cars or animals), of activity of neurons, etc. The high-dimensionality of these systems corresponds to the large number of variables (atoms, agents, neurons, respectively, in the preceding examples) and typically makes these systems very expensive to simulate and hard to understand even when simulations are possible. This research project aims to develop novel ideas and algorithms for the model reduction of such systems. The investigator will develop automatic learning algorithms that, by collecting a number of short simulations in parallel, output a much lower-dimensional model of the original system that yields faster simulations, with provable accuracy, in a much reduced number of variables. This will make the simulations less expensive, allowing one to perform more and longer simulations, and make the extraction of useful information from simulated data easier. He will apply these constructions to molecular dynamics simulations, expecting to significantly reduce the computational time needed to simulate small biomolecules.Large data sets in high dimensional spaces appear in a wide variety of scientific fields and applications. The PI focuses here on data sets that originate from the simulation of high-dimensional stochastic systems that arise in a wide variety of applications (e.g. molecular dynamics), with the goal of producing a much lower dimensional stochastic system with similar statistical properties as the full system, at least at a certain time scale. This is possible for a wide variety of dynamical systems with separation of time scales when the structure of the forces acting on the system and the stochastic perturbations are such that the trajectories of the system accumulate, in state or phase space, around low-dimensional sets (at the appropriate time scale and accuracy). The approach only requires access to a simulator S for the original system that, given initial conditions and the shortest time scale of interest as inputs, produces as output a (stochastic) path of the system starting at that initial condition and stops at the specified time. A call to S is typically expensive, but after a small number of carefully designed calls that yield a relatively small number of short paths, the algorithm learns and outputs a low-dimensional representation of the system, that is, a low-dimensional stochastic system whose trajectories are (in a suitable statistical sense), at the requested time scale, close to those of the original system. This construction may be performed in an online setting, as new regions of state space are explored, and in a multiscale fashion, where the time scale at which the system is reduced varies. These techniques will be adaptive to the assumed low intrinsic dimension of the simulated data, the timescale of interest, and the accuracy, leading to a new generation of results and algorithms for learning and approximating high-dimensional stochastic systems. While the techniques to be developed are applicable to a large family of stochastic systems, the main application considered in this project is Molecular Dynamics (MD). These techniques are expected to dramatically speed up the exploration of the state space of these molecules and of MD simulations as a whole. At the same time, they are general enough that they are applicable to a wide variety of stochastic systems, and the framework sets the stage for a novel approach to automatic learning of dynamical systems that is amenable to further generalizations.

高维随机动力系统出现在各种各样的科学领域和应用中，包括分子动力学模型，多主体系统（如汽车或动物），神经元活动等。这些系统的高维性对应于大量的变量（在前面的例子中分别是原子、代理、神经元），并且通常使这些系统的模拟成本非常高，即使可以进行模拟也很难理解。本研究项目旨在为这类系统的模型简化开发新的思路和算法。研究人员将开发自动学习算法，通过并行收集大量短模拟，输出原始系统的低维模型，该模型产生更快的模拟，具有可证明的准确性，变量数量大大减少。这将使模拟成本更低，允许执行更多和更长时间的模拟，并使从模拟数据中提取有用的信息更容易。他将把这些结构应用于分子动力学模拟，期望显著减少模拟小生物分子所需的计算时间。高维空间中的大数据集出现在各种科学领域和应用中。PI专注于源自高维随机系统模拟的数据集，这些系统出现在各种各样的应用中（例如分子动力学），其目标是产生一个具有与完整系统相似统计特性的低维随机系统，至少在一定的时间尺度上。当作用在系统上的力的结构和随机扰动使得系统的轨迹在低维集合周围（在适当的时间尺度和精度下）积累时，这对于具有分离时间尺度的各种动力系统是可能的。该方法只需要访问原始系统的模拟器S，给定初始条件和感兴趣的最短时间尺度作为输入，产生系统在该初始条件下开始并在指定时间停止的（随机）路径作为输出。对S的调用通常是昂贵的，但经过少量精心设计的调用，产生相对少量的短路径后，算法学习并输出系统的低维表示，即低维随机系统，其轨迹（在适当的统计意义上）在请求的时间尺度上接近原始系统的轨迹。当探索状态空间的新区域时，这种构建可以在在线环境中进行，并且以多尺度的方式进行，其中系统减少的时间尺度是不同的。这些技术将适应模拟数据的低固有维数、感兴趣的时间尺度和精度，从而产生新一代的结果和算法，用于学习和近似高维随机系统。虽然要开发的技术适用于大量随机系统，但本项目考虑的主要应用是分子动力学（MD）。这些技术有望大大加快这些分子的状态空间的探索和MD模拟作为一个整体。同时，它们具有足够的通用性，可以适用于各种各样的随机系统，并且该框架为动态系统自动学习的新方法奠定了基础，该方法可以进一步推广。