Probability & Statistics Seminar

Overcoming the curse of dimensionality: from nonlinear Monte Carlo to the training of neural networks

  • Speaker: Arnulf Jentzen (CUHK & University of Münster )

  • Time: Oct 14, 2022, 15:00-16:00

  • Location: Tencent Meeting ID 940-538-096

Abstract:PDEs are among the most universal tools used in modelling problems in nature and man-made complex systems. Nearly all traditional approximation algorithms for PDEs in the literature suffer from the so-called “curse of dimensionality” in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE. With such algorithms, it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used. In the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of dimensionality can be overcome by means of Monte Carlo approximation algorithms and the Feynman-Kac formula. In the first part of this talk, we present an efficient machine learning algorithm to approximate solutions of high-dimensional PDE and we also prove that suitable deep neural network approximations do indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs. In the second part of the talk we present some recent mathematical results on the training of neural networks.

Biography:Arnulf Jentzen is appointed as a presidential chair professor at the Chinese University of Hong Kong (since 2021) and as a full professor at the University of Münster (since 2019). The core research topics of his group are machine learning approximation algorithms, computational stochastics, numerical analysis for high dimensional PDEs, stochastic analysis, and computational finance. Currently he serves in the editorial boards of several scientific journals such as AAP, CMS, JML, SISC, and SINUM. He has been awarded the Felix Klein Prize from European Mathematical Society in 2020 and the Joseph F. Traub Prize for achievement in Information-Based Complexity in 2022.