Mathematical Sciences Department
Virtual Numerical Methods Seminar
Speaker: Yulong Lu, University of Minnesota, Twin Cities
Wednesday, November 1, 2023
2:00 pm - 3:00 pm
Zoom Meeting ID: 929 6655 1992
https://wpi.zoom.us/j/92966551992
Title: Machine learning meets PDEs
Abstract: Machine learning have recently been used to design innovative, and arguably revolutionary methods for solving many challenging problems from science and engineering which are modeled by partial differential equations (PDEs). Conversely, PDEs provide an important set of tools for understanding machine learning methods. This talk devotes to presenting some recent progress at the interface between neural networks-based machine learning and PDEs.
In the first part of the talk, we will discuss theoretical analysis of neural-network methods for solving high dimensional PDEs. We show that Deep Ritz solvers achieve dimension-free generalization rates in solving elliptic problems under the assumption that the solutions belong to Barron spaces. To justify such assumption, we develop new complexity-based solution theory for several elliptic problems in the Barron spaces.
In the second part of the talk, we will showcase the power of PDEs in minimax optimization, which underpins a variety of problems in adversarial machine learning. More precisely, we consider the problem of finding the mixed Nash equilibria (MNE) in two-player zero sum games on the space of probability measures. It is proved that two-scale gradient descent ascent (GDA) dynamics converges to the unique MNE of an entropy-regularized objective at an exponential rate. We also show that an annealed GDA with a logarithmically decaying cooling schedule converges to the MNE of the original unregularized objective.