Georgia Institute of Technology College of Sciences

Paper abstract: Recent advances in deep learning has witnessed many innovative frameworks that solve high dimensional mean-field games (MFG) accurately and efficiently. These methods, however, are restricted to solving single-instance MFG and demands extensive computational time per instance, limiting practicality. To overcome this, we develop a novel framework to learn the MFG solution operator. Our model takes a MFG instances as input and output their solutions with one forward pass. To ensure the proposed parametrization is well-suited for operator learning, we introduce and prove the notion of sampling invariance for our model, establishing its convergence to a continuous operator in the sampling limit. Our method features two key advantages. First, it is discretization-free, making it particularly suitable for learning operators of high-dimensional MFGs. Secondly, it can be trained without the need for access to supervised labels, significantly reducing the computational overhead associated with creating training datasets in existing operator learning methods. We test our framework on synthetic and realistic datasets with varying complexity and dimensionality to substantiate its robustness.

Link: https://arxiv.org/abs/2401.15482

I will give an overview of a research path in data driven modeling of complex systems over the last 30 or so years – from the early days of shallow neural networks and autoencoders for nonlinear dynamical system identification, to the more recent derivation of data driven “emergent” spaces in which to better learn generative PDE laws. In all illustrations presented, I will try to point out connections between the “traditional” numerical analysis we know and love, and the more modern data-driven tools and techniques we now have – and some mathematical questions they hopefully make possible for us to answer.

Bio: Yannis Kevrekidis studied Chemical Engineering at the National Technical University in Athens. He then followed the steps of many alumni of that department to the University of Minnesota, where he studied with Rutherford Aris and Lanny Schmidt (as well as Don Aronson and Dick McGehee in Math). He was a Director's Fellow at the Center for Nonlinear Studies in Los Alamos in 1985-86 (when the Soviet Union still existed and research funds were plentiful). He then had the good fortune of joining the faculty at Princeton, where he taught Chemical Engineering and also Applied and Computational Mathematics for 31 years; seven years ago he became Emeritus and started fresh at Johns Hopkins (where he somehow is also Professor of Urology). His work always had to do with nonlinear dynamics (from instabilities and bifurcation algorithms to spatiotemporal patterns to data science in the 90s, nonlinear identification, multiscale modeling, and back to data science/ML); and he had the additional good fortune to work with several truly talented experimentalists, like G. Ertl's group in Berlin. Currently -on leave from Hopkins- he works with the Defense Sciences Office at DARPA. When young and promising he was a Packard Fellow, a Presidential Young Investigator and the Ulam Scholar at Los Alamos National Laboratory. He holds the Colburn, CAST Wilhelm and Walker awards of the AIChE, the Crawford and the Reid prizes of SIAM, he is a member of the NAE, the American Academy of Arts and Sciences, and the Academy of Athens.

I'll talk about some specialized kirby calculus constructions: immersed surface complements and round handles. I'll prove using kirby calculus that S2xS2 minus an appropriate smooth embedded S2vS2 is diffeomorphic to R4. Maybe that is obvious, but the point is we can find nice diagrams where you see everything explicitly.

We discuss the sphere dimension of a graph. This is the smallest integer $d$ such that the graph can be represented as the intersection graph of a collection of spheres in $\mathbb{R}^d$. We show that graphs with small sphere dimension have small balanced separators, as long as they exclude a complete bipartite graph $K_{t,t}$. This property is connected to forbidding shallow minors and can be used to develop divide-and-conquer algorithms. This is joint work with James Davies, Agelos Georgakopoulos, and Meike Hatzel.