Numerical Analysis and Scientific Computing Seminar | Nat Trask, Structure preserving data-driven models via exterior calculus

Zoom (Please contact ddelreyfernandez@uwaterloo.ca for meeting link) 

Speaker

Nat Trask, Associate Professor, Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania

Title

Structure preserving data-driven models via exterior calculus

Abstract

Despite the recent flurry of work employing machine learning to develop surrogate models to accelerate scientific computation, the "black-box" underpinnings of current techniques fail to provide the verification and validation guarantees provided by modern finite element methods. In this talk we present a data-driven finite element exterior calculus for developing reduced-order models of multiphysics systems when the governing equations are either unknown or require closure. The framework employs deep learning architectures typically used for logistic classification to construct a trainable partition of unity which provides notions of control volumes with associated boundary operators. This alternative to a traditional finite element mesh is fully differentiable and allows construction of a discrete de Rham complex with a corresponding Hodge theory. We demonstrate how models may be obtained with the same robustness guarantees as traditional mixed finite element discretization, with deep connections to contemporary techniques in graph neural networks. For applications developing digital twins where surrogates are intended to support real time data assimilation and optimal control, we further develop the framework to support Bayesian optimization of unknown physics on the underlying adjacency matrices of the chain complex. By framing the learning of fluxes via an optimal recovery problem with a computationally tractable posterior distribution, we are able to develop models with intrinsic representations of epistemic uncertainty. Finally, we demonstrate extensions of this work to dynamical systems, where we may learn dynamics while preserving by construction notions of energy conservation, entropy production, and fluctuation-dissipation balance.