Past Event: Oden Institute Seminar

Geometric Methods for the Approximation of High-dimensional Dynamical Systems

Mauro Maggioni, Bloomberg Distinguished Professor of Mathematics, and Applied Mathematics & Statistics, Johns Hopkins University

Abstract

We discuss geometry-based statistical learning techniques for performing model reduction and modeling of certain classes of stochastic high-dimensional dynamical systems. We consider two complementary settings. In the first one, we are given long trajectories of a system, e.g. from molecular dynamics, and we estimate, in a robust fashion, an effective number of degrees of freedom of the system, which may vary in the state space of the system, and a local scale where the dynamics is well-approximated by a reduced dynamics with a small number of degrees of freedom. We then use these ideas to produce an approximation to the generator of the system and obtain, via eigenfunctions of an empirical Fokker-Planck equation (constructed from data), reaction coordinates for the system that capture the large time behavior of the dynamics. We present various examples from molecular dynamics illustrating these ideas. In the second setting we only have access to a (large number of expensive) simulators that can return short paths of the stochastic system, and introduce a statistical learning framework for estimating local approximations to the system, that can be (automatically) pieced together to form a fast global reduced model for the system, called ATLAS. ATLAS is guaranteed to be accurate (in the sense of producing stochastic paths whose distribution is close to that of paths generated by the original system) not only at small time scales, but also at large time scales, under suitable assumptions on the dynamics. We discuss applications to homogenization of rough diffusions in low and high dimensions, as well as relatively simple systems with separations of time scales, and deterministic chaotic systems in high dimensions, that are well-approximated by stochastic diffusion-like equations. Bio Dr. Mauro Maggioni is a Bloomberg Distinguished Professor of Mathematics, and Applied Mathematics and Statistics at Johns Hopkins University. He works at the intersection of harmonic analysis, approximation theory, high-dimensional probability, statistical and machine learning, model reduction, stochastic dynamical systems, spectral graph theory, and statistical signal processing. He received his B.Sc. in Mathematics summa cum laude at the Universitá degli Studi in Milano in 1999, and the Ph.D. in Mathematics from the Washington University, St. Louis, in 2002. He was a Gibbs Assistant Professor in Mathematics at Yale University till 2006, when he moved to Duke University, becoming Professor in Mathematics, Electrical and Computer Engineering, and Computer Science in 2013. He received the Popov Prize in Approximation Theory in 2007, an NSF CAREER award and Sloan Fellowship in 2008, and was nominated Fellow of the American Mathematical Society in 2013.