AARMS Scientific Machine Learning Seminar: Ben Adcock (Simon Fraser University)

Title: Tackling the curse: polynomial and deep neural network methods for function approximation in high dimensions

Abstract: Many problems in computational science and engineering require the accurate approximation of a target function from data. This problem is rendered challenging by the high-dimensionality of the function, the expense of generating function samples, the presence of noise in the measurements, and the fact that the target function may take values in a function space. Developing techniques that tackle these challenges without succumbing to the famous “curse of dimensionality” has been a long-standing problem. In the first part of this talk I will give a brief survey of a decade’s worth of progress on high-dimensional function approximation via sparse polynomial expansions. I will show how the proper use of nonlinear approximation theory and compressed sensing techniques leads to algorithms for high-dimensional approximation which, unlike other approaches, possess provably near-optimal error bounds and moderate sample complexities. In particular, these techniques mitigate the curse of dimensionality to a substantial degree. The second part of the talk will be devoted to emerging approaches based on deep neural networks and deep learning. Such tools are beginning to garner substantial attention in the scientific computing community. Nonetheless, I will present evidence of a key gap between current theory and practice. I will then discuss recent results showing that there exist deep neural networks that match the performance of best-in-class schemes, and furthermore, these can indeed be trained through realizable procedures. This highlights the potential of deep neural networks, and sheds light on achieving robust, reliable and overall improved practical performance.

This talk is based on joint work with Anyi Bao, Simone Brugiapaglia, Juan M. Cardenas, Nick Dexter, Sebastian Moraga, Yi Sui and Clayton G. Webster.

Meeting link:
https://mun.webex.com/mun/j.php?MTID=m38cc6ce370de2b1c234e33feac5bb93a

Meeting number: 2630 760 7617
Password: cJwPcrHM425