Applied and Computational Mathematics Seminars

COSAM Departments Mathematics & Statistics Research Seminars Applied and Computational Mathematics



Upcoming Applied and Computational Mathematics Seminars
DMS Applied and Computational Mathematics Seminar
Apr 11, 2025 02:00 PM
328 Parker Hall


 
 jamesscott
 
Dr. James Scott  (incoming faculty member in Applied Mathematics)
 
Title: Nonlocal Boundary Value Problems with Local Boundary Conditions
 
 
Abstract: We state and analyze nonlocal problems with classically-defined, local boundary conditions. The model takes its horizon parameter to be spatially dependent, vanishing near the boundary of the domain. We establish a Green's identity for the nonlocal operator that recovers the classical boundary integral, which permits the use of variational techniques. Using this, we show the existence of weak solutions, as well as their variational convergence to classical counterparts as the bulk horizon parameter uniformly converges to zero. In certain circumstances, global regularity of solutions can be established, resulting in improved modes and rates of variational convergence. Generalizations of these results pertaining to models in continuum mechanics and Laplacian learning will also be presented.
 
 

More Events...

Past Applied and Computational Mathematics Seminars
DMS Applied and Computational Mathematics Seminar
Apr 04, 2025 02:00 PM
328 Parker Hall


moleitao

Speaker: Molei Tao (Georgia Tech)  
 
Title: Optimization, Sampling, and Generative Modeling in Non-Euclidean Spaces


Abstract: Machine learning in non-Euclidean spaces have been rapidly attracting attention in recent years, and this talk will give some examples of progress on its mathematical and algorithmic foundations. A sequence of developments that eventually leads to the generative modeling of data on Lie groups will be reported. Such a problem occurs, for example, in the Gen-AI design of molecules.

More precisely, I will begin with variational optimization, which, together with delicate interplays between continuous- and discrete-time dynamics, enables the construction of momentum-accelerated algorithms that optimize functions defined on manifolds. Selected applications, such as a generic improvement of Transformer, and a low-dim. approximation of high-dim. optimal transport distance, will be described. Then I will turn the optimization dynamics into an algorithm that samples from probability distributions on Lie groups. This sampler provably converges, even without log-concavity condition or its common relaxations. Finally, I will describe how this sampler can lead to a structurally-pleasant diffusion generative model that allows users to, given training data that follow any latent statistical distribution on a Lie group manifold, generate more data exactly on the same manifold that follow the same distribution. If time permits, applications such as molecule design and generative innovation of quantum processes will be briefly discussed.

Short bio: 
Molei Tao is a full professor in School of Math at Georgia Tech, working on the mathematical foundations of machine learning. He received B.S. from Tsinghua Univ. and Ph.D. from Caltech, and worked as a Courant Instructor at NYU before starting at Georgia Tech. He serves as an Area Chair for NeurIPS, ICLR and ICML, and he is a recipient of W.P. Carey Ph.D. Prize in Applied Mathematics (2011), American Control Conference Best Student Paper Finalist (2013), NSF CAREER Award (2019), AISTATS best paper award (2020), IEEE EFTF-IFCS Best Student Paper Finalist (2021), Cullen-Peck Scholar Award (2022), GT-Emory AI.Humanity Award (2023), SONY Faculty Innovation Award (2024), Best Poster Award at an international conference “Recent Advances and Future Directions for Sampling” held at Yale (2024), as well as several other recognitions.
 
 
DMS Applied and Computational Mathematics Seminar
Mar 28, 2025 02:00 PM
328 Parker Hall


zhong
 
Speaker: Dr. Yimin Zhong (Auburn)
 
Title: Numerical Understanding of Neural Networks
 
 
Abstract: In this talk, I will talk about a couple of recent works on neural networks. The motivation is to see whether neural networks are suitable for general scientific computing. Our study of shallow neural networks demonstrates that shallow neural networks are in general low-pass filters from different perspectives. Based on this observation, we proposed to make use of the composition of shallow networks to construct deep neural networks, which demonstrates better performance over the vanilla fully connected neural networks of comparable parameters.
 
 
This talk is a part of Dr. Zhong's 3rd year review process.
 
ACM seminar’s website: https://auburn.edu/cosam/mathseminar/index.htm
 
DMS Applied and Computational Mathematics Seminar
Mar 21, 2025 02:00 PM
328 Parker Hall


qitang
 
Speaker: Qi Tang (Georgia Tech) 
 
Title: Structure-preserving machine learning for learning dynamical systems
 
 
Abstract: I will present our recent work on structure-preserving machine learning (ML) for dynamical systems. First, I introduce a structure-preserving neural ODE framework that accurately captures chaotic dynamics in dissipative systems. Inspired by the inertial manifold theorem, our model learns the ODE’s right-hand side by combining a linear and a nonlinear term, enabling long-term stability on the attractor for the Kuramoto-Sivashinsky equation. This framework is further enhanced with exponential integrators. Next, I discuss ML for singularly perturbed systems, leveraging the Fenichel normal form to simplify fast dynamics near slow manifolds. A fast-slow neural network is proposed that enforces the existence of a trainable, attractive invariant slow manifold as a hard constraint.
 
DMS Applied and Computational Mathematics Seminar
Feb 21, 2025 02:00 PM
328 Parker Hall


weizhu

Speaker: Wei Zhu (Georgia Tech)  

Title: Symmetry-Preserving Machine Learning: Theory and Applications

 

Abstract: Symmetry underlies many machine learning and scientific computing tasks, from computer vision to physical system modeling. Models designed to respect symmetry often perform better, but several questions remain. How can we measure and maintain approximate symmetry when real-world symmetries are imperfect? How much training data can symmetry-based models save? And in non-convex optimization, do these models truly converge to better solutions? In this talk, I will share my work on these challenges, revealing that the answers are sometimes surprising. The approach draws on applied probability, harmonic analysis, differential geometry, and optimization, but no specialized background is required.

DMS Applied and Computational Mathematics Seminar
Feb 14, 2025 02:00 PM
328 Parker Hall


zhong
 
Speaker: Yimin Zhong (Auburn)
 
Title: Fast solvers for radiative transfer and beyond
 
 
Abstract:  Despite the tremendous developments in recent years, constructing efficient numerical solution methods for the radiative transfer equation (RTE) is still challenging in scientific computing. In this talk, I will present a simple yet fast computational algorithm for solving the RTE in isotropic media in steady-state and time-dependent settings. The algorithm we developed has two steps. In the first step, we solve a volume integral equation for the angularly averaged solution using iterative schemes such as the GMRES method. The computation in this step is accelerated with a variant of the fast multipole method (FMM). In the second step, we solve a scattering-free transport equation to recover the angular dependence of the solution. The algorithm does not require the underlying medium to be homogeneous. We present numerical simulations under various scenarios to demonstrate the performance of the proposed numerical algorithm for both homogeneous and heterogeneous media. Then I will extend the formulation to the time-domain and anisotropic scattering media and analyze the possibility of applying the fast algorithm. 
DMS Applied and Computational Mathematics Seminar
Nov 22, 2024 01:00 PM
328 Parker Hall


liu

Speaker: Yi Liu (Auburn University)

Title: Convergence Analysis of the ADAM Algorithm for Linear Inverse Problems

 

Abstract:  The ADAM algorithm is one of the most popular stochastic optimization methods in machine learning. Its remarkable performance in training models with massive datasets suggests its potential efficiency in solving large-scale inverse problems. In this work, we apply the ADAM algorithm to solve linear inverse problems and establish the sub-exponential convergence rate for the algorithm when the noise is absent. Based on the convergence analysis, we present an a priori stopping criterion for the ADAM iteration when applied to solve inverse problems at the presence of noise. The convergence analysis is achieved via the construction of suitable Lyapunov functions for the algorithm when it is viewed as a dynamical system with respect to the iteration numbers. At each iteration, we establish the error estimates for the iterated solutions by analyzing the constructed Lyapunov functions via stochastic analysis. Various numerical examples are conducted to support the theoretical findings and to compare with the performance of the stochastic gradient descent (SGD) method. 

DMS Applied and Computational Mathematics Seminar
Nov 15, 2024 02:00 PM
ZOOM


 
Speaker: Patrizio Bifulco (FernUniversität in Hagen, Germany)
 
Title: Comparing the spectrum of Schrodinger operators on metric graphs using heat kernels
 
 
Abstract: We study Schrodinger operators on compact finite metric graphs subject to \(\delta\)-coupling and standard boundary conditions often known as Kirchoff-Neumann vertex conditions. We compare the \(n\)-th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the mean value of the eigenvalue deviations which represents a generalization to a recent result by Rudnick, Wigman and Yesha obtained for domains in \(\mathbb{R}^2\) to the setting of metric graphs. We start this talk by introducing the basic notion of a metric graph and discuss some basic properties of heat kernels on those graphs afterwards. In this way, we are able to discuss a so-called local Weyl law which is relevant for the proof of the asymptotic main result. If time permits, we will also briefly discuss the case of \(\delta'\)-coupling conditions and some possible generalizations on infinite graphs having finite total length.
 
This talk is based on joint works with Joachim Kerner (Hagen) and Delio Mugnolo (Hagen).
DMS Applied and Computational Mathematics Seminar
Nov 08, 2024 02:00 PM
328 Parker Hall


watson

Speaker: Alexander Watson (University of Minnesota Twin Cities)

Title: Multiple-scales perspective on moiré materials 

 

Abstract: In recent years, experiments have shown that twisted bilayer graphene and other so-called "moiré materials" realize a variety of important strongly-correlated electronic phases, such as superconductivity and fractional quantum anomalous Hall states. I will present a rigorous multiple-scales analysis justifying the (single-particle) Bistritzer-MacDonald PDE model, which played a critical role in the prediction of these phases in twisted bilayer graphene. The significance of this model is that it has moiré-periodic coefficients even when the underlying material is aperiodic at the atomic scale. This allows moiré materials to be studied via Floquet-Bloch band theory, a variant of the Fourier transform. I will then discuss generalizations of this model and other mathematical questions related to moiré materials.

DMS Applied and Computational Mathematics Seminar
Oct 29, 2024 02:00 PM
228 Parker Hall


deang

Speaker: Dr. Jennifer Deang (Lockheed Martin; affiliated faculty member of DMS) 

Title: On the Mathematical Perspective of the Missile Defense System

 

Abstract: We first provide an overview of the systems, weapons, and technology needed for detection, tracking, interception, and destruction of attacking missiles. Then we will outline the current research areas sought by the MDA to advance and solve complex technological problems, ultimately contributing to a more robust Missile Defense System (MDS). 

DMS Applied and Computational Mathematics Seminar
Oct 25, 2024 01:00 PM
328 Parker Hall


musslimani

Speaker: Ziad Musslimani (Florida State University)

Title: Space-time nonlocal integrable systems

 

Abstract: In this talk I will review past and recent results pertaining to the emerging topic of integrable space-time nonlocal integrable nonlinear evolution equations. In particular, we will discuss blow-up in finite time for solitons and the physical derivations of many integrable nonlocal systems.

More Events...