The Department of Applied Mathematics weekly seminar is given by scholars and researchers working in applied mathematics, broadly interpreted.
Title: A Low Rank Neural Representation of Nonlinear Shock Waves
Abstract: Despite decades of significant progress in scientific computing, real-time solvers of parametrized partial differential equations (pPDEs) describing nonlinear wave phenomena generally remains out of reach. One important obstacle is the apparent lack of low-dimensional structure in wave phenomena. For example, it is known that the Kolmogorov width decays slowly for solution manifolds of nonlinear conservation laws, prohibiting construction of efficient reduced basis methods. In this talk, I will discuss theoretical and computational results that point towards a new low-dimensional structure in convective problems, as described by a neural network architecture we call low rank neural representation (LRNR), that can potentially lead to new dimensionality reduction tools as well as to new real-time solvers applicable to various pPDEs. In particular, I will discuss how (1) a theoretical LRNR construction with small parameter dimension can efficiently approximate solutions to scalar nonlinear conservation laws involving arbitrarily complicated shock interactions, (2) LRNRs can be used within the physics informed neural networks (PINNs) to efficiently compute pPDE solutions.
This is talk is based on joint works with Woojin Cho (Yonsei U.), Kookjin Lee (Arizona State U.), Noseong Park (KAIST), Gerrit Welper (U. Central Florida), and Randall J. LeVeque (U. Washington).