A priori generalization error analysis of neural network methods for solving high dimensional elliptic PDEs

Add to your list(s) Download to your calendar using vCal

• Yulong Lu (University of Massachusetts)
• Tuesday 29 June 2021, 15:00-16:00
• https://bham-ac-uk.zoom.us/j/3900445385?pwd=Mk1LSmRiUk5STEJMWVl0bE82NTM1QT09.

If you have a question about this talk, please contact Hong Duong.

Neural network-based machine learning methods, including the most notably deep learning have achieved extraordinary successes in numerous fields. Despite the rapid development of learning algorithms based on neural networks, their mathematical analysis is far from understood. In particular, it has been a big mystery that neural network-based machine learning methods work extremely well for solving high dimensional problems.

In this talk, we will demonstrate the power of neural network methods for solving high dimensional elliptic PDEs. Specifically, we will discuss an a priori generalization error analysis of the Deep Ritz Method for solving two classes of high dimensional Schrödinger problems: the stationary Schrödinger equation and the ground state of Schrödinger operator. Assuming the exact solution or the ground state lies in a low-complexity function space called spectral Barron space, we show that the convergence rate of the generalization error is independent of dimension. We also develop a new regularity theory for the PDEs of consideration on the spectral Barron space. This can be viewed as an analog of the classical Sobolev regularity theory for PDEs.

This talk is part of the Data Science and Computational Statistics Seminar series.

This talk is included in these lists:

• Data Science and Computational Statistics Seminar
• School of Mathematics events
• https://bham-ac-uk.zoom.us/j/3900445385?pwd=Mk1LSmRiUk5STEJMWVl0bE82NTM1QT09

Note that ex-directory lists are not shown.