ATLAS: A Geometric Approach to Learning High-Dimensional Stochastic Systems Near Manifolds

Abstract

When simulating multiscale stochastic differential equations (SDEs) in high-dimensions, separation of timescales, stochastic noise and high-dimensionality can make simulations prohibitively expensive. The computational cost is dictated by microscale properties and interactions of many variables, while the behavior of interest often occurs at the macroscale level and at large time scales, often characterized by few important, but unknown, degrees of freedom. For many problems bridging the gap between the microscale and macroscale by direct simulation is computationally infeasible. In this work we propose a novel approach to automatically learn a reduced model with an associated fast macroscale simulator. Our unsupervised learning algorithm uses short parallelizable microscale simulations to learn provably accurate macroscale SDE models, which are continuous in space and time. The learning algorithm takes as input: the microscale simulator, a local distance function, and a homogenization spatial or temporal scale, which is the smallest time scale of interest in the reduced system. The learned macroscale model can then be used for fast computation and storage of long simulations. We prove guarantees that related the number of short paths requested from the microscale simulator to the accuracy of the learned macroscale simulator. We discuss various examples, both low-and high-dimensional, as well as results about the accuracy of the fast simulators we construct, and its dependency on the number of short paths requested from the microscale simulator.

Data-driven spectral decomposition and forecasting of ergodic dynamical systems

“…The vast array of methods developed to address these goals can be broadly categorized as state-space or operator-theoretic methods [1]. A popular state-space approach is to approximate a nonlinear dynamical system by a collection of local linear models on the tangent planes of the attractor [2][3][4]; other approaches construct global nonlinear models for the dynamical evolution map [5], or nonlinearly project the attractor to lower-dimensional Euclidean spaces and construct reduced models operating in those spaces [6][7][8]. A common element of these techniques is that the forward operators of the reduced models are defined in state space; i.e., they map the state at a given time to another state in the future.…”

Learning Interaction Kernels in Stochastic Systems of Interacting Particles from Multiple Trajectories

“…In this work, we assume that the noise coefficient is a known constant: there has been of course significant work in estimating the noise coefficient, for example in the case of interacting particle systems see the recent work [26] and references therein, and for the case of model reduction for Langevin equations with state-dependent diffusion coefficient [19].…”

“…For deterministic multi-particle systems, various types of learning techniques have been developed (see, e.g., [9,14,40,41,50,55] and the reference therein). When it comes to stochastic multi-particle systems, only a few efforts have been made, e.g., learning reduced Langevin equations on manifolds in [19] (without, however, assuming nor exploiting the structure of pairwise interactions), learning parametric potential functions in [10,15] from single trajectory data, estimating the diffusion parameter in [26], and estimating effective Langevin equations on manifolds in [19].…”

Transition Manifolds of Complex Metastable Systems

“… 2006 ; Singer et al. 2009 ; Crosskey and Maggioni 2017 ; Vanden-Eijnden 2007 ; Kevrekidis and Samaey 2009 ). However, all of these approaches still depend on some local form of time scale separation between the “fast” and the “slow” components of the dynamics.…”