Speaker: Donsub Rim, Washington University
Title: Low Rank Neural Representation of Hyperbolic Conservation Laws
Abstract: A Low Rank Neural Representation (LRNR) is a parametrized family of feedforward neural networks whose weights and biases belong to low rank linear subspaces. In this talk, we will discuss how LRNRs can serve as efficient low dimensional representations of solutions to hyperbolic conservation laws. First, we motivate the LRNR architecture by reformulating the entropy solutions to scalar conservation laws to reveal their low dimensional structure. Next, we will show that LRNRs can be trained from numerical solution data through a meta-learning approach, and demonstrate how the trained LRNRs possess important properties: (1) low dimensionality, (2) smoothness and stability with respect to the parameters even in the presence of shocks, and (3) its ability to backpropagate with complexity scaling with the low dimension only. Its applications in the popular Physics Informed Neural Networks (PINNs) framework will also be discussed.
This talk is based on joint works with Woojin Cho (Yonsei U.), Kookjin Lee (Arizona State U.), Noseong Park (KAIST), and Gerrit Welper (U. Central Florida).
Bio: Donsub Rim is an assistant professor in the Department of Mathematics and Statistics at Washington University. Rim’s research interests include numerical analysis of partial differential equations and inverse problems. His current projects range from nonlinear model reduction methods using deep neural networks to computational applications in aerospace engineering, geophysics, and medical imaging, including rocket combustion dynamics, probabilistic tsunami hazard assessment, storm surge prediction, and coupled-physics imaging. Rim earned his doctorate in applied mathematics at the University of Washington, Seattle. Most recently, Rim was a postdoctoral research associate at the Courant Institute of Mathematical Sciences at New York University and a Chu Assistant Professor at Columbia University.
In person attendance at this seminar is only open to Columbia Univesity affiliates. External guests are welcome to attend remotely. Please contact [email protected] if you need the Zoom link for this seminar.