CSE PhD Thesis Defense: Emily Williams

Abstract:

Discretizations of many multiscale systems representing complex physical phenomena contain too many degrees of freedom to simulate accurately given limited computational resources. A motivating example is turbulent flow, which is present in many engineering and scientific applications. Large-eddy simulation has demonstrated capabilities in modeling turbulent flow at a more affordable computational cost by filtering the Navier-Stokes equations. This coarse-graining introduces subgrid-scale (SGS) stress terms that represent the effect of the unresolved motions on the resolved flow and must be modeled. Model reduction is another strategy toward computationally efficient simulation. Projection-based reduced-order models (ROMs) aim to solve the governing dynamics on significantly fewer degrees of freedom, by projecting the equations onto a basis of reduced dimensionality compared to the original solution space.

Traditional deterministic SGS models can be overly dissipative or unstable, especially in regions of turbulent flow. Ongoing work in earth systems modeling motivates the use of stochastic SGS models for chaotic dynamics. An accurate SGS model is essential because unresolved dynamics can significantly impact the degrees of freedom that are kept through memory terms, arising from the Mori-Zwanzig (MZ) formalism from nonequilibrium statistical mechanics.

In this thesis, we study stochastic and generative modeling approaches for complex dynamical systems governed by multiscale and chaotic differential equations, with a particular motivation of turbulent flow. We represent the coarse-scale dynamics as stochastic differential equations (SDEs) and then propose probabilistic closure models using generative modeling via stochastic integration, leading to improved predictions of resolved variables. Further, we demonstrate the stabilizing effect that these stochastic models have on linearized chaotic dynamics, assessed through forward sensitivity analysis. We also use the MZ formalism to build ROMs for basis functions obtained through physics-informed machine learning. The fluctuation-dissipation theorem is implemented to construct non-Markovian evolution equations for the expansion coefficients of the basis functions. In addition to MZ, we use a generative approach to develop closure models. Both for MZ and the generative approaches, we obtain improved predictions of the filtered states over the Markovian dynamics alone.

Thesis Committee Members:

• Prof. David L. Darmofal, Jerome B. Wiesner Professor of Aeronautics and Astronautics, MIT
• Prof. Themistoklis P. Sapsis, William I. Koch Professor of Mechanical Engineering, MIT
• Dr. Panagiotis Stinis, Computational Mathematics Group, Pacific Northwest National Laboratory
• Dr. Amanda A. Howard, Computational Mathematics Group, Pacific Northwest National Laboratory
• Dr. Patrick J. Blonigan, Principal Member of Technical Staff, Sandia National Laboratories (Reader)
• Dr. Keaton J. Burns, Principal Research Scientist of Mathematics, MIT (Reader)