Spring 2025 MMAE Seminar Series: Zack Hilliard

The Department of Mechanical, Materials, and Aerospace Engineering presents its spring 2025 seminar series featuring Zack Hilliard, a Ph.D. candidate at North Carolina State University. Hilliard will present “Sequential Data Assimilation for PDEs Using Shape-Morphing Solutions.” This seminar is open to the public and will take place on Wednesday, January 22, 2025, from 12:45–1:45 p.m. in room 104 of the Rettaliata Engineering Center.

Abstract

Shape-morphing solutions (also known as evolutional deep neural networks, reduced-order non-linear solutions, and neural Galerkin schemes) are a new class of methods for approximating the solution of time-dependent partial differential equations (PDEs). In this talk, we introduce a sequential data assimilation method for incorporating observational data in a shape-morphing solution (SMS). Our method takes the form of a predictor-corrector scheme, where the observations are used to correct the SMS parameters using Newton-like iterations. Between observation points, the SMS equations—a set of ordinary differential equations—are used to evolve the solution forward in time. We demonstrate the efficacy of DA-SMS on three examples: the nonlinear Schrödinger equation, the Kuramoto–Sivashinsky equation, and a two-dimensional advection-diffusion equation. Our numerical results suggest that DA-SMS converges with relatively sparse observations and a single iteration of the Newton-like method.

Biography

Originally from South Carolina, Zack Hilliard went out West to complete a bachelor’s degree in mathematics at University of California, Santa Barbara. Upon completion, he moved back to the East Coast to pursue a master’s degree in mathematical sciences at the College of Charleston. Directly after, he started a Ph.D. at North Carolina State University. Under the supervision of Professor Mohammad Farazmand, he is expected to receive his Ph.D. later this year in the summer 2025.