MAC-MIGS Talks by Prof Richard Tsai – Thursday 7th April 2022

Yen-Hsi Richard Tsai is a professor of mathematics at the University of Texas at Austin and a fellow of the Oden Institute for Computational Engineering and Sciences. His work is in scientific computing and he has made contributions to machine learning, multiscale modelling, numerical analysis, robotic navigation and medical imaging. He has held a Sloan Fellowship, a Simons Fellowship and obtained a number of other prizes and awards.

On Thursday April 7th, we plan to host two talks by Professor Richard Tsai. This event will be in person only. Places are limited and preference will be given to students and staff involved with MAC-MIGS.

Organisers: Benedict Leimkuhler and Peter Whalley

Timetable

Abstracts

Side-effects of Learning from Low Dimensional Data Embedded in an Euclidean Space

The low dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low dimensional manifolds embedded in a high dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as input. However, one often needs to consider evaluating the optimized network at points outside the training distribution. We analyze the cases where the training data are distributed in a linear subspace of Rd. We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace. We study the potential regularization effects associated with the network’s depth and noise in the codimension of the data manifold.

Time-parallel computation of Hamiltonian systems aided by machine learning

We propose a machine learning approach for enhancing time-parallel computation of Hamiltonian systems. We will demonstrate the approach by computing wave propagation in media with multiscale wave speed, using a second-order linear wave equation model as a proof-of-concept. We advocate the use of online- and offline-data for enhancing the parareal algorithm of Lions, Turinici and Maday, and demonstrate that the coupled approach improves the stability of parareal algorithms for wave propagation and improves the accuracy of the enhanced coarse solvers. We use neural networks to exploit data collected offline and a novel Procrustes approach for data collected “online” during the parareal iterations. We discuss the generation of suitable training data and compare the performance of networks trained by different data sets. We find that the approach can be effective as long as the causality in the given system is appropriately sampled in training data.