Upcoming Event: Oden Institute Seminar

Numerics Informed Neural Networks for Solving PDEs

Beatrice Riviere, Rice University

Abstract

Due to the popularity of AI, there is an increased interest in applying machine learning algorithms to model physical phenomena in science and engineering. However, purely data driven methods fail to capture accurate solutions for problems that rely on physical conservation laws; these problems are characterized by coupled and nonlinear partial differential equations. The field of scientific machine learning (SciML) addresses the drawbacks of purely data driven machine learning by constraining the learned solution to physical laws. In this talk, we formulate  two SciML methods that learn a finite-dimensional representation of the true solution of the PDE. In the first approach, the learned solution is shown to be close to the finite difference solution whereas in the second approach, the  method is inspired by the discontinuous Galerkin method. Our numerics informed neural network (NINN) method is mesh aware and utilizes knowledge from classical numerical methods for PDEs. In both approaches, we show convergence of the NINN methods as the mesh size tends to zero under the assumption of a controlled optimization error. Various numerical examples confirm the theoretical error bounds.

Biography

Beatrice Riviere is a Noah Harding Chair and Professor in the Department of Computational and Applied Mathematics and Operations Research and a member of the Ken Kennedy Institute at Rice University. She has worked extensively on the formulation and analysis of numerical methods applied to problems in porous media and in fluid mechanics. She is the author of more than one hundred and fifty scientific publications in numerical analysis and scientific computation. She published two books with SIAM: one on the theory and implementation of discontinuous Galerkin methods that is highly cited and one that appeared in 2024 on the mathematics and finite elements for Navier-Stokes. Dr. Riviere is a SIAM Fellow (2021), an AWM Fellow (2022) and an IACM Fellow (2024). She currently serves as Chair of the SIAM Board of Trustees. She was elected President of the SIAM TX-LA Section from 2020 to 2022 and Chair of the SIAM Activity Group on Geosciences from 2019 to 2020. She leads a research cluster in Scientific Machine Learning in the Ken Kennedy Institute at Rice. Dr. Riviere’s research group, COMP-M, has been funded by the National Science Foundation, the National Institutes of Health, the oil and gas industry and the Gulf Coast Consortia for the Quantitative Biomedical Sciences.