Connections to Machine Learning $$ \nonumber \newcommand{\xb}{\boldsymbol{x}} \newcommand{\yb}{\boldsymbol{y}} \newcommand{\zb}{\boldsymbol{z}} \newcommand{\thetab}{\boldsymbol{\theta}} \newcommand{\grad}{\nabla} \newcommand{\RR}{\mathbb{R}} $$ In my previous post, I introduced the notion of proximal gradient descent and explained the way in which the “geometry” or the metric used in the proximal scheme allows us to define gradient flows on arbitrary metric spaces. This concept is important in the context of statistical mechanics because analysis of the Fokker-Planck equation naturally yields a gradient flow in the Wasserstein metric.

Wasserstein Gradient Flows and the Fokker Planck Equation (Part I) $$ \nonumber \newcommand{\xb}{\boldsymbol{x}} \newcommand{\yb}{\boldsymbol{y}} \newcommand{\grad}{\nabla} \newcommand{\RR}{\mathbb{R}} $$ The connection between partial differential equations arising in chemical physics, like the Fokker-Planck equation discussed below, and the notions of distance in the space of probability measures is a relatively young set of mathematical ideas. While the theory of gradient flows of arbitrary metric spaces can get exceedingly intricate, the fundamental ideas are not unapproachable.

Machine learning (ML) is spurring a transformation in the computational sciences by providing a new way to build flexible, universal, and efficient approximations for complex high-dimensional functions and functionals. One area in which the impact of these new tools is beginning to be understood is the physical sciences, where traditionally intractable high-dimensional partial differential equations are now within reach. This tutorial will explore how developments in ML complement computational problems in the physical sciences, with a particular focus on solving partial differential equations, where the challenges of high-dimensionality and data acquisition also arise. The first important example this tutorial will cover is using Deep Learning Methods for solving high-dimensional PDEs, which have wide application in variational rare events calculations, many-body quantum systems, and stochastic control. Another challenge covered by this tutorial that researchers often face is the complexity or lack of specification of the models they are using when performing uncertainty quantification. Thus another line of research aims to recover the underlying dynamic using observational data. This tutorial will introduce the well-developed methods and theories for using machine learning in scientific computing. We will first discuss how to incorporate physical priors into machine learning models. Next, we will discuss how these methods can help to solve challenging physical and chemical problems. Finally, we will discuss the statistical and computational theory for scientific machine learning. In this tutorial, we will not focus on the technical details behind these theories, but on how they can help the audience to understand the challenges of using machine learning in differential equation applications and to develop new methods for addressing these challenges.

The goal of emulating biology has stimulated many investigations into properties that lead to robust assembly. Many of the works that focus on designing self-assembly have pursued a strategy of tuning the interactions among the various components to stabilize a given target structure. In this talk, I will discuss some work that approaches this problem from a distinct viewpoint, namely, through the lens of nonequilibrium control processes in which external perturbations are tuned to drive the assembly dynamics to states unreachable in equilibrium. I will discuss computational strategies for carrying out an optimization to control the steady state of interacting particle systems using only external fields. In addition, I will introduce a framework to quantify the dissipative costs of maintaining a nonequilibrium steady state that uses observable properties alone.

In probability theory, the notion of weak convergence is often used to describe two equivalent probability distributions. This metric requires equivalence of the average value of well-behaved functions under the two probability distributions being compared. In coarse-grained modeling, Noid and Voth developed a thermodynamic equivalence principle that has a similar requirement. Nevertheless, there are many functions of the fine-grained system that we simply cannot evaluate on the coarse-grained degrees of freedom. In this talk, I will describe an approach that combines accelerated sampling of a coarse-grained model with invertible neural networks to invert a coarse-graining map in a statistically precise fashion. I will show that for non-trivial biomolecular systems, we can recover the fine-grained free energy surface from coarse-grained sampling.