51²è¹Ý

Spring, 2022

Department of Mathematics, 51²è¹Ý
Room: 126 Clements Hall (if not specified, Refreshment starts 15 minutes before talks)

 

[1] Speaker: Prof. Lu Lu, Dept of Chemical and Biomolecular Engineering, U. Penn, February 18

Time: 3:45pm, Friday, Feb. 18, 2022

Title:  Learning nonlinear operators using deep neural networks for diverse applications

 

Abstract: It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators or complex systems from streams of scattered data. In this talk, I will present the deep operator network (DeepONet) to learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations. More generally, DeepONet can learn multiscale operators spanning across many scales and trained by diverse sources of data simultaneously. I will demonstrate the effectiveness of DeepONet to multiphysics and multiscale problems.

 

Biosketch: Lu Lu is an Assistant Professor in the Department of Chemical and Biomolecular Engineering at University of Pennsylvania. He is also a faculty in Penn Institute for Computational Science. Prior to joining Penn, he was an Applied Mathematics Instructor in the Department of Mathematics at Massachusetts Institute of Technology from 2020 to 2021. He obtained his Ph.D. degree in Applied Mathematics at Brown University in 2020, master's degrees in Engineering, Applied Mathematics, and Computer Science at Brown University, and bachelor's degrees in Mechanical Engineering, Economics, and Computer Science at Tsinghua University in 2013. Lu has a multidisciplinary research background with research experience at the interface of applied mathematics, physics, computational biology, and computer science. The goal of his research is to model and simulate physical and biological systems at different scales by integrating modeling, simulation, and machine learning, and to provide strategies for system learning, prediction, optimization, and decision making in real time. His current research interest lies in scientific machine learning, including theory, algorithms, and software, and its applications to engineering, physical, and biological problems. His broad research interests focus on multiscale modeling and high performance computing for physical and biological systems. Lu has received the 2020 Joukowsky Family Foundation Outstanding Dissertation Award of Brown University.

 

 

[2] Speaker: Prof. Yuan Gao, Dept of Math, Purdue University, March 3, 2022

 

Time: 3:45pm, Thursday, Mar. 3, 2022

 

Title: Droplets with moving contact line and insoluble surfactant: Onsager’s principle, variational inequality, computations

 

 

ABSTRACT.   The capillary effect caused by the interfacial energy dominates the dynamics of small droplets, particularly the contact lines (where three phases meet). With insoluble surfactant laid on the capillary surface, the adhesion of droplets to some textured substrates becomes more complicated:  (i) insoluble surfactant disperses along  the evolving capillary surface  (ii) the surfactant-dependent surface tension will in turn drive the full dynamics of droplets, particularly the moving contact lines. Using Onsager’s principle with different Rayleigh dissipation functionals, we derive and compare both the geometric motion of the droplets and the viscous flow model  with Marangoni effect. To enforce impermeable obstacle constraint, the full dynamics of the droplet can be formulated as a gradient flow on a manifold with boundary, and two equivalent variational inequalities are derived. We propose unconditionally stable first/second order numerical schemes based on explicit moving boundaries and arbitrary-Lagrangian Eulerian method.  After adapting a projection method for the variational inequality with phase transition information at emerged contact lines, we compute the contact angle hysteresis, unavoidable splitting/merging of droplets on inclined textured substrates.

 

Biosketch:  Dr. Yuan Gao is an assistant professor in the Department of Mathematics at Purdue University. She was a William W. Elliott Assistant Research Professor at the Department of Mathematics, Duke University during 2019-2021. Before joining Duke, she received her PhD from Fudan University in 2017 and was a Postdoc at Hong Kong University of Science and Technology. Yuan Gao specializes in calculus of variation and numerical analysis for singular nonlinear PDEs rising from crystalline materials, image sciences and fluid dynamics. She also works on applied stochastic analysis and algorithms for multiscale dynamics and optimal control for the transition path theory in non-equilibrium biochemical reactions.

 

 

[3] Speaker: Prof. Jianguo Liu, Dept of Math, Duke University, March 4

 

Time: 11a-noon, Friday, Mar. 4, 2022

 

Title: Macroscopic dynamics for nonequilibrium biochemical reactions from a Hamiltonian viewpoint

  

Abstract:  Most biochemical reactions in living cells are open system interacting with environment through chemostats. At a mesoscopic scale, the number of  each species in those biochemical reactions can be modeled by the random time-changed Poisson processes. To characterize the macroscopic behaviors in the large volume limit, the law of large number in path space determines a mean-field limit nonlinear Kurtz ODE, while the WKB expansion yields a Hamilton-Jacobi equation and the corresponding Lagrangian gives the good rate function in the large deviation principle. A parametric variation principle can be formulated to compute the reaction paths. We propose a gauge-symmetry criteria for a class of non-equilibrium chemical reactions including   enzyme reactions, which identifies a new concept of balance within the same reaction vector due to flux grouping degeneracy. With this criteria,  we (i) formulate an Onsager-type gradient flow structure in terms of the energy landscape given by a steady solution to the Hamilton-Jacobi equation;  (ii) find transition paths between multiple non-equilibrium steady states (rare events in biochemical reactions).  We illustrate this idea through a bistable catalysis  reaction. In contrast to the standard diffusion approximations via Kramers-Moyal expansion, a new drift-diffusion approximation sharing the same gauge-symmetry is constructed based on the Onsager-type gradient flow formulation to compute the correct energy barrier. 

 

Biosketch:   Jian-Guo Liu earned his BS and PhD from Fudan University and UCLA, respectively. He was an Courant Instructor at NYU before joining the Department of Mathematics at Temple University, then he moved to University of Maryland, College Park, and then joined to Duke University as a professor of Mathematics and Physics. Dr Liu’s research is in the areas of PDE, numerical analysis and applied mathematics in general. His current research interests includes crystal growth, surface dynamics, water wave systems, analysis of machine learning algorithms, interacting particle systems, non-equilibrium chemical reactions, analysis some biological systems, integrable system. He is a fellow of AMS.

 

 

[4] Speaker: Prof. Bjorn Engquist, Dept of Math, UT Austin, March 31, 2022

 

Time: 3:45pm, Thursday, Mar. 31, 2022

 

Title: Stochastic gradient descent algorithm for global optimization

 

Abstract: We propose an adaptive stochastic gradient descent algorithm for finding the global minimum of a non-convex objective function f(x). By using the value of the function f(x) for tuning the randomness, convergence can be proved. The rate is algebraic in the number of iterates both in probability and in the x-space. This is a substantial improvement over the classical logarithmic convergence rate. The analysis is focused on the case where f(x) is deterministic and the randomness added for global convergence. We will remark on problems where the objective function is intrinsically random or random after sampling as often in machine learning. Numerical examples will be presented.

 

Bio-sketch: Bjorn Engquist received his Ph.D. in numerical analysis from Uppsala University in 1975. He has been Professor of Mathematics at UCLA, and the Michael Henry Stater University Professor at Princeton University. He was director of the Princeton Program in Applied and Computational Mathematics and the Princeton Institute for Computational Science. Engquist came to The University of Texas at Austin in 2004, where he is Professor of Mathematics holding the Computational and Applied Mathematics Chair I, and is Director of the Oden Institute Center for Numerical Analysis. Engquist is a member of the Royal Swedish Academy of Sciences, the Royal Swedish Academy of Engineering Sciences and the Norwegian Academy of Science and Letters. He is also a member of the American Academy of Arts and Sciences and a SIAM Fellow. His honors include Guggenheim fellowship, Henrici and Birkhoff Prizes and Celsius and College de France medals.

Engquist’s research focuses on development, analysis and application of numerical methods for differential equations. Examples of topics are absorbing boundary conditions, forward and inverse problems of wave propagation, algorithms for compressible flow, homogenization and multi-scale modeling.

 

 

 

[5] Speaker:  Prof. Stas Molchanov, Math, UNC Charlotte, April 14, 2022 on zoom.

 

Time: 3:45pm, Thursday, April 14, 2022

 

Title:  Dynamo - Theorem (review)

 

Abstract. Dynamo – theorem describes the generation of magnetic field by the random (turbulent) motion of the conducting fluid. It was introduced to explain the existence and the nature of the black spots on the sun. The evolution of the solar magnetic field can be represented by the Maxwell equation with the short-correlated velocity field of plasma and the corresponding matrix potential. The central result of the theory is the existence of the temperature - induced phase transition: the exponential growth of the field is possible only for high enough temperature. In this case, the magnetic field highly non-homogeneous (intermittent).

The talk will present recent and not very recent results in this area.

 

Bio-sketch:      From 1958 to 1963 he was a student at the Mathematical and Mechanical faculty, Moscow State University (MSU), where he graduated in 1963 with a master degree supervised by Eugene Dynkin. At MSU Molchanov graduated in 1967 with Russian Candidate degree (Ph.D.)  and in 1983 with Russian Doctor of Sciences degree (higher doctoral degree). At MSU he was from 1966 to 1971 an assistant professor, from 1971 to 1988 an associate professor, and from 1988 to 1990 a full professor in the department of probability theory and mathematical statistics. He was a visiting professor from 1991 to 1992 at the University of California, Irvine and from 1992 to 1993 at the University of Southern California. In 1994 Molchanov became a full professor at the University of North Carolina at Charlotte.

 

His research deals with geometrical approaches to Markov processes (Martin boundaries and diffusion on Riemannian manifolds) and with spectral theory (localization in random media and spectral properties of Riemannian manifolds). His research on applied mathematics includes physical processes and fields in disordered structures involving averaging and intermittency with applications to geophysics, astrophysics, oceanography. With regard to physical processes, he has done research on wave processes in periodic and random media, quantum graphs, and applications to optics.

 

With Ilya Goldsheid and Leonid Pastur he proved in 1977 localization in the Anderson model in one dimension. With Michael Aizenman, Molchanov proved in 1993 localization for large coupling constants and energies near the edge of the spectrum. 

 

In 1990 he was an invited speaker at the International Congress of Mathematicians in Kyoto.  In 2012 he became a Fellow of the American Mathematical Society.