Applied Mathematics Research Seminar by George Karniadakis: Research Seminar: The Mathematics of Neural PDEs and Neural Operators

Speaker: George Karniadakis, Charles Pitts Robinson and John Palmer Barstow Professor of Applied Mathematics at Brown University

Title: Research Seminar: The Mathematics of Neural PDEs and Neural Operators, Mies Campus Classroom

Abstract: Physics-informed neural networks (PINNs) and neural operators like DeepOnet have formed
the foundations of Scientific Machine Learning (SciML) with applications across all domains in science,
engineering and biomedicine. Here, will present a mathematical view of these neural networks, analyzing
their convergence, and provide interpretations for their successes and failures in some prototype
benchmark problems but also realistic applications. Specifically, we will discuss how PINNs can be used
to tackle the curse-of-dimensionality, and how DeepOnet can be used to obtain in real-time 3D flow fields
for a large distribution of parameters. We will also discuss the limitations of these methods for predicting
long-term dynamics even for simple problems for which conventional methods succeed.