Abstract:
We discuss a geometry-based statistical learning framework for
performing
model reduction and modeling 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 discuss new
techniques for estimating, 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 question, 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 simulations of
high-dimensional stochastic system, and introduce a novel statistical
learning
framework for learning automatically a family of 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, stochastic or chaotic,
that are
well-approximated by stochastic differential equations. |