*A Theoretical Laboratory for Mesoscale Biophysics*

Biology is influenced by processes from the quantum to the macroscopic scale, a fundamental challenge to studying biophysical dynamics. We study the time-evolution of biological systems at *mesoscales*, where the molecular meets the macroscopic.

We use techniques from statistical inference and machine learning to sample, coarsegrain, and interpret complex chemical and biophysical models. We also characterize and analyze numerical methods to guarantee accuracy and robustness.

We study the self-organization of biophysical and chemical species as dictate by intrinsic material properties. Additionally, we are interested in analyzing self-assembly in nonequilibrium conditions as well as exploiting external control to dictate the outcomes of an assembly process.

Notes on research and other topics of interest

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.

In many applications in computational physics and chemistry, we seek to estimate expectation values of observables that yield mechanistic insight about reactions, transitions, and other “rare” events. These problems are often plagued by metastability; slow relaxation between metastable basins leads to slow convergence of estimators of such expectations. In this talk, I will focus on efforts to exploit developments in generative modeling to sample distributions that are challenging to sample with local dynamics (e.g., MCMC or molecular dynamics) due to metastability. I will discuss the problem of sampling when there is not a large, pre-existing data set on which to train. By simultaneously sampling with traditional methods and learning a sampler, we assess the prospects of neural network driven sampling to accelerate convergence and to aid exploration of high-dimensional distributions. This is joint work with Marylou Gabrié and Eric Vanden-Eijnden.

A pedagogical overview of the mean-field approach to studying the dynamics and trainability of neural networks.

The surprising flexibility and undeniable empirical success of machine learning algorithms has inspired many theoretical explanations for the efficacy of neural networks. Here, I will briefly introduce one perspective that provides not only asymptotic guarantees of trainability and accuracy in high-dimensional learning problems, but also provides some prescriptions and design principles for learning. Bolstered by the favorable scaling of these algorithms in high dimensional problems, I will turn to a central problem in computational condensed matter physics—that of computing reaction pathways. From the perspective of an applied mathematician, these problems typically appear hopeless; they are not only high-dimensional, but also dominated by rare events. However, with neural networks in the toolkit, at least the dimensionality is somewhat less intimidating. I will describe an algorithm that combines stochastic gradient descent with importance sampling to optimize a function representation of a reaction pathway for an arbitrary system. Finally, I will provide numerical evidence of the power and limitations of this approach.

Controlling thermodynamic cycles to minimize the dissipated heat is a longstanding goal in thermodynamics, and more recently, a central …

Sampling the collective, dynamical fluctuations that lead to nonequilibrium pattern formation requires probing rare regions of …

Sampling the collective, dynamical fluctuations that lead to nonequilibrium pattern formation requires probing rare regions of …

Self-assembly, the process by which interacting components form well-defined and often intricate structures, is typically thought of as …

Self-assembly, the process by which interacting components form well-defined and often intricate structures, is typically thought of as …

Many problems in the physical sciences, machine learning, and statistical inference necessitate sampling from a high-dimensional, …

Recent theoretical work has characterized the dynamics of wide shallow neural networks trained via gradient descent in an asymptotic …

Deep neural networks, when optimized with sufficient data, provide accurate representations of high-dimensional functions; in contrast, …