Georgia Institute of Technology College of Sciences

Leveraging large-scale data and systems of computing accelerators, statistical learning has led to  significant paradigm shifts in many scientific disciplines. Grand challenges in science have been tackled with exciting synergy between disciplinary science, physics-based simulations via high-performance computing, and powerful learning methods.

In this talk, I will describe several vignettes of our research in the theme of modeling complex dynamical systems characterized by partial differential equations with turbulent solutions. I will also demonstrate how machine learning technologies, especially advances in generative AI technology,  are effectively applied to address the computational and modeling challenges in such systems, exemplified by their successful applications to  weather forecast and climate projection. I will also discuss what new challenges and opportunities have been brought into future machine learning research.

The research work presented in this talk is based on joint and interdisciplinary research work of several teams at Google Research, ETH and Caltech.


Bio: Dr. Fei Sha is currently a research scientist at Google Research, where he leads a team of scientists and engineers working on scientific machine learning with a specific application focus towards AI for Weather and Climate. He was a full professor and the Zohrab A. Kaprielian Fellow in Engineering at the Department of Computer Science, University of Southern California. His primary research interests are machine learning and its application to various AI problems: speech and language processing, computer vision, robotics and recently scientific computing, dynamical systems, weather forecast and climate modeling.  Dr. Sha was selected as a Alfred P. Sloan Research Fellow in 2013, and also won an Army Research Office Young Investigator Award in 2012. He has a Ph.D from Computer and Information Science from U. of Pennsylvania and B.Sc and M.Sc from Southeast University (Nanjing, China). More information about Dr. Sha's scholastic activities can be found at his microsite at http://feisha.org.

Given a finite set of points $\Gamma$ in $\mathbb{P}^n$, we say that $\Gamma$ satisfies the Cayley-Bacharach condition with respect to degree r polynomials, or is CB(r), if any degree r homogeneous polynomial F vanishing on all but one point of $\Gamma$ must vanish at the last point. In recent literature, the condition has played an important role in computing a birational invariant called the degree of irrationality of complex projective varieties. However, the condition itself has not been studied extensively, and surprisingly little is known about the geometric properties of CB(r) points. 

In this talk, I will discuss a new combinatorial approach to the study of the CB(r) condition, using matroid theory, and present some examples of how matroid theory can shed light on the underlying geometry of such sets.
 

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.