Thursday at 2:00-3:00 PM (unless otherwise noted)
Hybrid format: either virtually or in Math Tower Room 154 (See below for details)
For questions, contact Dr. Yulong Xing or Dr. Dongbin Xiu, Email: xing dot 205@osu.edu or xiu dot 16@osu.edu
| DATE and TIME | Location | SPEAKER | TITLE | December 2
Thursday, 2pm |
ZOOM ID: 967 9869 9124
Pascode: OSUMath |
Houman Owhadi
(Caltech) |
Computational Graph Completion |
| January 20
Thursday, 2pm |
ZOOM ID: 991 1883 3389
Pascode: OSUMath |
Lexing Ying
(Stanford) |
Provably convergent quasistatic dynamics for mean-field two-player zero-sum games | March 3
Thursday, 2pm |
In person
Math Tower 154 |
J. Nathan Kutz
(Univ. of Washington) |
The Future of Governing Equations |
We introduce a framework for generating, organizing, and reasoning with computational knowledge. It is motivated by the observation that most problems in Computational Sciences and Engineering (CSE) can be described as that of completing (from data) a computational graph representing dependencies between functions and variables. Functions and variables may be known, unknown, or random. Data comes in the form of observations of distinct values of a finite number of subsets of the variables of the graph (satisfying its functional dependencies). The underlying problem combines a regression problem (approximating unknown functions) with a matrix completion problem (recovering unobserved variables in the data). Replacing unknown functions by Gaussian Processes (GPs) and conditioning on observed data provides a simple but efficient approach to completing such graphs. Since the proposed framework is highly expressive, it has a vast potential application scope. Since the completion process can be automatized, as one solves $\sqrt{\sqrt{2}+\sqrt{3}}$ on a pocket calculator without thinking about it, one could, with the proposed framework, solve a complex CSE problem by drawing a diagram. Compared to traditional regression/kriging, the proposed framework can be used to recover unknown functions with much scarcer data by exploiting interdependencies between multiple functions and variables. Numerous examples illustrate the scope and efficacy of the proposed framework and show how it can be used as a pathway to identifying simple solutions to classical CSE problems. These examples include the seamless recovery of known methods (for solving/learning PDEs, surrogate/multiscale modeling, mode decomposition, deep learning) and the discovery of new ones (digital twin modeling, dimension reduction).
We study the minimax problem arising from finding the mixed Nash equilibrium for mean-field two-player zero-sum games. Solving this problem requires optimizing over two probability distributions. We consider a quasistatic Wasserstein gradient flow dynamics in which one probability distribution follows the Wasserstein gradient flow, while the other one is always at the equilibrium. Theoretical analysis are conducted on this dynamics, showing its convergence to the mixed Nash equilibrium under mild conditions. Inspired by the continuous dynamics of probability distributions, we derive a quasistatic Langevin gradient descent method with inner-outer iterations, and test the method on different problems, including training mixture of GANs.
Machine learning and AI algorithms are transforming a diverse number of fields in science and engineering. This is largely due their success in model discovery which turns data into reduced order models and neural network representations that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems, compact representations, and their embeddings from data. Importantly, data-driven architectures must jointly discover coordinates and parsimonious models in order to produce maximally generalizable and interpretable models of physics-based systems and processes.
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