Events

Maseeh Mathematics & Statistics Colloquium: Learning and Geometry for Stochastic Dynamical Systems in high dimensions
Friday, April 13, 2018 - 3:15pm

The Maseeh Mathematics and Statistics Colloquium Series*
presents


Mauro Maggioni, Ph.D.
Johns Hopkins University

Learning and Geometry for Stochastic Dynamical Systems in high dimensions


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 then 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.

Friday, April 13, 2018 at 3:15pm
Urban Center Building room 304
(Refreshments served at 2:45PM in same room)

* Sponsored by the Maseeh Mathematics and Statistics Colloquium Series Fund and the Fariborz Maseeh Department of Mathematics & Statistics, PSU. This event is free and open to the public.