Opportunities for the deep neural network method of solving partial differential equations in the computational study of biomolecules driven through periodic geometries

Abstract

As deep learning emerged in the 2010s to become a groundbreaking technology in machine vision and natural language processing, it also ushered in many new algorithms for use in scientific research. Among these is the neural network method, in which the solution to a differential equation is approximated by varying the parameters of a deep neural network trial function. Although this idea has been explored with shallow neural networks since the 1990s, it has experienced a resurgence of interest in recent years now that it can be implemented with deep neural networks. A series of empirical and theoretical studies have acclaimed the deep variants of the neural network method for being able to solve many classes of traditionally challenging partial differential equations. These early works emphasized its potential to solve high-dimensional, highly parameterized, and nonlinear equations in arbitrary geometries, all without requiring the discretization of the geometry into a mesh. Problems exhibiting these challenging features abound in computational biophysics, and this thesis presents recent efforts to adapt the neural network method for use in this _eld. The investigations in this thesis center on models of biomolecular motion in periodic geometries. Such models arise, for example, in the study of microfluidic and nanofluidic devices used for the separation of free-draining molecules. These problems exhibit many of the characteristics for which the neural network method is appealing, and serve here as non-trivial test problems on which to characterize its performance. Perspectives from biophysics, numerical analysis, and deep learning are combined to elucidate the true potential of the neural network method as a technique for studying such differential equations. Altogether, these works have moved the neural network method closer to being another reliable numerical method in the computational biophysicist's toolkit.