Statistical Learning for High-Dimensional Stochastic Dynamical Systems

高维随机动力系统出现在广泛的科学领域和应用中，包括分子的动力学模型、多智能体系统(如汽车或动物)的模型、神经元活动的模型等。这些系统的高维对应于大量的变量(在前面的例子中分别是原子、智能体、神经元)，并且通常使得这些系统的模拟非常昂贵，即使在模拟可能的情况下也很难理解。该研究项目旨在为这类系统的模型降阶开发新的思想和算法。研究人员将开发自动学习算法，通过并行收集大量短期模拟，输出原始系统的低维模型，该模型在变量数量大大减少的情况下产生更快的模拟，并具有可证明的准确性。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信息变得更容易。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广泛的科学领域和应用中。PI在这里关注的是来自于在广泛的应用(例如分子动力学)中出现的高维随机系统的模拟的数据集，其目标是产生至少在特定时间尺度上具有与完整系统相似的统计特性的低得多的维随机系统。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状态或相空间中围绕低维集合(以适当的时间尺度和精度)累积时，这对于具有时间尺度分离的各种动力系统是可能的。该方法只需要访问原始系统的仿真器S，该仿真器以给定的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开始到指定时刻停止的(随机)路径作为输出。对S的调用通常是昂贵的，但在产生相对较少数量的短路径的少量精心设计的调用之后，该算法学习并输出系统的低维表示，即其轨迹(在适当的统计意义上)在所请求的时间尺度上接近于原始系统的轨迹的低维随机系统。当探索状态空间的新区域时，该构造可以在在线设置中执行，并且可以以多尺度方式来执行，其中系统减小的时间尺度是不同的。这些技术将适应假设的模拟数据的低本征维、感兴趣的时间尺度和精度，导致新一代结果和算法用于学习和逼近高维随机系统。虽然将要开发的技术适用于一大类随机系统，但本项目考虑的主要应用是分子动力学(MD)。这些技术有望极大地加快探索这些分子的状态空间和整个MD模拟的速度。同时，它们足够普遍，适用于各种各样的随机系统，并且该框架为动态系统的自动学习的新方法奠定了基础，该方法易于进一步推广。

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.