Mathematics Colloquium
Deep Learning Meets Optimal Control and Dynamical Systems: A Mathematical Approach
When
February 6, 2023 | 3:30 p.m. - 4:30 p.m.
Refreshments at 3:00 in Lobby
Where
University Hall 1005
Speaker
Deepanshu Verma
Abstract
Deep neural networks, or DNNs, have become a popular tool in various fields due to their universal approximation properties. This talk delves into the intricacies of designing efficient DNN algorithms and architectures, as well as exploring their use in solving optimal control problems. DNN architectures are known to be difficult to design and train, often resulting in challenges such as the exploding or vanishing gradients issue. These networks also exhibit a large number of equivalent optimal solutions in the parameter space. A neural network architecture designed using fractional derivatives will be introduced to tackle the former challenge. Additionally, the notion of bias ordering, with theoretical guarantees, will be introduced to narrow down the parameter search space. The second half of the talk will focus on the applications of DNNs in high-dimensional optimal control problems. Traditional methods for solving these problems often suffer from the Curse-of-Dimensionality, where computational complexity increases exponentially with the dimension of the problem. Utilizing DNNs to approximate the value function of control problems can effectively tackle this issue. One of the key challenges in training is discovering the relevant parts of the state space. To address this, techniques from control theory will be employed to devise a self-supervised training algorithm. Several numerical experiments, including applications to PDE constrained optimization, will be presented.
Novel Data-driven Techniques in Reduced-Order Modeling of Fluid Flows
When
February 3, 2023 | 3:30 p.m. - 4:30 p.m.
Refreshments at 3:00 in Lobby
Where
University Hall 1005
Speaker
Dr. Muhammad Mohebujjaman
Abstract
In this talk, we present a data-driven filtered reduced order model (DDF-ROM) framework for the numerical simulation of fluid flows. The novel DDF-ROM framework consists of two steps: (i) In the first step, we use ROM projection to filter the nonlinear PDE and construct a filtered ROM. This filtered ROM is low-dimensional but is not closed. (ii) In the second step, we use data-driven modeling to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved modes. At the end, we enforce physical constraints in the data-driven modeling of the ROM closure term. The constrained data-driven ROM is significantly more accurate than its unconstrained counterpart. Finally, we present a novel high-order ROM differential filter and use it in conjunction with an evolve-filter-relax algorithm to attenuate the numerical oscillations of standard ROMs. We also examine how stochastic collocation methods can be combined with the EFR algorithm for efficient UQ of fluid flows.
Uniqueness and L2 stability of compressible Euler equations
When
December 2, 2022 | 2:30 p.m. - 3:30 p.m.
Refreshments to follow
Where
University Hall 4002
Speaker
Geng Chen
Abstract
Compressible Euler equations are a typical system of hyperbolic conservation laws, whose solution forms shock waves in general. It is well known that global BV solutions of system of hyperbolic conservation laws exist, when one considers small BV initial data. In this talk, we will present our recent proof on uniqueness of BV solution. This is a joint work with Sam Krupa and Alexis Vasseur.
How much further is it? A stroll through metric and length space
When
November 16, 2022 | 2:30 p.m. - 3:30 p.m.
Refreshments to follow
Where
University Hall 4002
Speaker
Peter Stollmann
Abstract
We review a simple dual characterization of length spaces, metric spaces in which points can be joined by a fairly short path.
On a conjecture by Erdoes, Pollack and Tuza
When
October 21, 2022 | 2:30 p.m. - 3:30 p.m.
Refreshments to follow
Where
University Hall 4002
Speaker
Eva Czabarka
Abstract
The Krein Extension: Basics and some concrete applications
When
October 12, 2022 | Reception: 3:30 p.m. | Lecture: 4:00 p.m.
Where
University Hall 1005
Speaker
Dr. Fritz Gesztesy, Baylor University
Description
2022 Roger T. Lewis Endowed Lectureship in Mathematics. Presented by the UAB Department of Mathematics.
