Abstract: A major obstacle in the deployment of spectral methods is the choice of appropriate bases for trial and test spaces. If chosen suitably, these basis functions lead invariably to well-posed discretized problems and well-conditioned linear systems, while the resulting approximate solutions are provably high-order accurate. However, barring domain decomposition approaches, devising such functions for arbitrary geometries from scratch is a hugely challenging task. Fortunately, the recently developed DeepONet approach is a highly promising device for generating machine-learned basis functions. In this talk, we propose a Galerkin approach for time-dependent partial differential equations that is powered by basis functions gleaned from the DeepONet architecture. We shall outline our procedure of obtaining these basis functions and detail their many favorable properties. Next, we shall present the results of numerical tests for various problems, including advection, advection-diffusion, and the viscous Burgers’ equations, as well as some highly intriguing preliminary results from the low-viscosity regime. Finally, we will identify potential pitfalls in generalizing to higher dimensions and suggest possible remedies.
Panos Stinis specializes in scientific computing with application interests in model reduction of complex systems, multiscale modeling, uncertainty quantification, and machine learning. Stinis studied aeronautical engineering at the Technical University of Athens, Greece. He earned his PhD in applied mathematics in 2003, from Columbia University in New York, in the area of model reduction. He began his career at Lawrence Berkeley National Laboratory and the Stanford Center for Turbulence Research, where he worked on applying model reduction methods to hyperbolic systems and in developing techniques for locating and tracking singularities of partial differential equations. In 2008, he became a faculty member at the Mathematics Department at the University of Minnesota, where he worked on renormalization, mesh refinement, particle filtering and optimization. He moved to the Pacific Northwest National Laboratory in 2014, where he is currently leading the Computational Mathematics group.