Paul Atzberger, Professor, Department of Mathematics, University of California, Santa Barbara
Recent emerging data-driven methods combined with more traditional numerical analysis are presenting new opportunities for model development and for performing simulations. We will discuss a few motivating applications in fluid mechanics and biophysics. We first discuss challenges in biophysical modeling of membrane proteins arising from the roles played by geometry and transport equations on curved surfaces. We discuss development of hybrid data-driven solvers for partial differential equations on manifolds. We show how these methods can be used to study membrane protein interactions and drift-diffusion dynamics taking into account the roles of hydrodynamic coupling, geometry, and thermal fluctuations. We then discuss how representations can be learned for non-linear stochastic dynamics leveraging recent data-driven methods related to Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs). We show how these methods can be used to develop reduced-order models, dimension reductions, or learn unknown force-laws. We present results for partial differential equations in fluid mechanics, reaction-diffusion processes, and particle systems. Throughout, we aim to highlight opportunities for combining recent emerging machine learning methods with more traditional numerical approaches to develop practical computational methods for scientific modeling and simulation.