**Fall 2022**

**Room 126 Clements Hall, Wednesday 3:45-4:45p; Refreshment starts 15 minutes before the talk**

**[1] Speaker: Prof. Dongbin Xiu, Department of Mathematics, OSU, September 14, 2022**

**Title**: Data Driven Modeling of Unknown Systems with Deep Neural Networks

**Abstract**: We present a framework of predictive modeling of unknown system from measurement data. The method is designed to discover/approximate the unknown evolution operator behind the data. Deep neural network (DNN) is employed to construct such an approximation. Once an accurate DNN model for evolution operator is constructed, it serves as a predictive model for the unknown system and enables us to conduct system analysis. We demonstrate that residual network (ResNet) is particularly suitable for modeling autonomous dynamical systems. Extensions to other types of systems will be discussed, including non-autonomous systems, systems with uncertain parameters, and more importantly, systems with missing variables, as well as partial differential equations

(PDEs).

**Biosketch : **Dongbin Xiu received his Ph.D degree from Division of Applied Mathematics of Brown University in 2004. He conducted post doctoral studies in Los Alamos National Laboratory, Princeton University, and Brown University, before joining the

Department of Mathematics of Purdue University as an Assistant Professor in the fall of 2005. He was promoted to the rank of Associate Professor in 2009 and to Full Professor in 2012. In 2013, he moved to the University of Utah as a Professor in the Department of Mathematics and Scientific Computing and Imaging (SCI) Institute. In 2016, He moved to The Ohio State University as Professor of Mathematics and Ohio Eminent Scholar. He has received NSF CAREER award in 2007, as well as a number of teaching awards at Purdue. He has served on the editorial board of several journals, including SIAM Journal on Scientific Computing and Journal of Computational Physics. He is the founding Associate Editor-in-Chief of the International Journal for Uncertainty Quantification (IJUQ), and the founding Editor-in-Chief of Journal of Machine Learning for Modeling and Computing (JMLMC). His research focuses on developing efficient numerical algorithms for uncertainty quantification, stochastic computing, and machine learning

**[2] Speaker: Prof. Haomin Zhou, Department of Mathematics, Gatech, October 5, 2022**

**Title: **Inverse Weak Adversarial Networks (iWAN): A Computational Method for High-dimensional Inverse Problems

**ABSTRACT**. In this talk, I will present a weak adversarial network approach to solve a class of inverse problems. Using the weak formulation of PDE, we rewrite the inverse problem as a minimax problem. Leveraged with deep neural networks, the solution of inverse problem, including the solution of PDE and the unknown media, can be solved simultaneously by finding the network parameters for the saddle point. While the parameters are updated, the networks gradually approximate the solution of the inverse problem. Theoretical justifications are provided on the convergence of the proposed algorithm. The proposed method is mesh-free without any spatial discretization and is suitable for problems with high dimensionality and low regularity on solutions. Numerical experiments on a variety of test problems demonstrate the promising accuracy and efficiency of this approach. This presentation is based on the joint work with Gang Bao (Zhejiang U.), Xiaojing Ye (Georgia State U.) and Yaohua Zang (Zhejiang U.).

**Biosketch: **Haomin Zhou is a professor in the School of Mathematics at Georgia Institute of Technology. He received his B.S. in pure mathematics from Peking University, M.Phil in applied mathematics from the Chinese University of Hong Kong, and Ph.D. in applied mathematics from University of California, Los Angeles in 1991, 1996 and 2000 respectively. He spent 3 years in California Institute of Technology as a postdoctoral scholar and von Karman instructor, before joining Georgia Institute of Technology as an assistant professor in 2003. His research interests includes numerical analysis and scientific computing, specialized in PDE and wavelet techniques in image processing, numerical methods for stochastic differential equations, and discrete optimal transport. He is a recipient of the NSF CAREER AWARD in applied and computational mathematics in 2007, and Feng Kang prize in scientific computing in 2019.

**[3] Speaker: Prof. Chun Liu, Department of Mathematics, IIT, Duke University, October 19, 2022**

**Title: Energetic variational approaches: dynamic boundary conditions
and thermal effects**

**Abstract: ** In this talk, I will present a multiscal/multiphysics energetic variational approaches for a wide class of dynamical system that is relevant in biological and engineering applications. In particular, I will present recent work on the evolution of grain boundaries. These are joint work with Yekaterina Epshteyn and Masashi Mizuno, and is partially supported by a NSF-DMREF award.

**Biosketch:** Chun Liu is the Chair and Professor in the Department of Applied Mathematics in Illinois Institute of Technology in Chicago. Before coming to Illinois Tech, Liu was in the Department of Mathematics at Pennsylvania State University, where he had served since 1998. He also served a term as associate director for the Institute for Mathematics and Its Applications (IMA) at the University of Minnesota, and has held positions at many institutions, such as the University of Wuerzburg, the University of Tokyo, the University of Georgia, and Carnegie Mellon University. He received his Ph.D. in 1995 from the Courant Institute of Mathematical Sciences at New York University.

Liu’s research is in nonlinear partial differential equations and applications in complex fluids, such as liquid crystal growth, polymers, ion channels in cell membranes,

and active materials involving chemical reactions. He developed a general framework of energetic variational approaches (EnVarA) to study various problems arising from physical and biological applications. Liu’s research has been supported by various federal and international funding agencies..

**[4] Speaker: Prof. A. Bensoussan, Department of Mathematics, UT Dallas, October 26, 2022**

**Title: ** **CONTROL ON HILBERT SPACES AND MEAN FIELD CONTROL**

**Abstract**: In this work, we describe an alternative approach to the general theory of Mean Field Control as presented in the book of P. Cardaliaguet, F. Delarue, J-M Lasry, P-L Lions: The Master Equation and the Convergence Problem in Mean Field Games, Annals of Mathematical Studies, Princeton University Press, 2019. Since it uses Control Theory and not P.D.E. techniques it applies only to Mean Field Control. The general difficulty of Mean Field Control is that the state of the dynamic system is a probability. Therefore, the natural

functional space for the state is the Wasserstein metric space. P.L. Lions has suggested to use the correspondence between probability measures and random variables, so that the Wasserstein metric space is replaced with the Hilbert space of square integrable random variables. This idea is called the lifting approach. Unfortunately, this brilliant idea meets some difficulties, which prevents to use it as an alternative, except in particular cases. In using a different Hilbert space, we study a Control problem with state in a Hilbert space, which solves the original Mean Field Control problem, as a particular case, and thus provides a complete alternative to the

approach of Cardaliaguet, Delarue, Lasry, Lions.

**Bio-sketch**: **Alain Bensoussan **is Lars Magnus Ericsson Chair and the Director of ICDRiA (International Center for Decision and Risk Analysis) at the University of Texas at Dallas He is also Chair Professor of Risk and Decision Analysis at the City University Hong Kong. He has been for 4 years World Class University Distinguished Professor at Ajou University, Korea . He is Professor Emeritus at the University Paris Dauphine. Professor Bensoussan served as President of National Institute for Research in Computer Science and Control (INRIA) from 1984 to 1996; President of the French Space Agency (CNES) from 1996 to 2003; and Chairman of the European Space Agency (ESA) Council from 1999 to 2002. He is a member of the French Academy of Sciences, French Academy of Technology, Academia Europaea, and International Academy of Astronautics. His distinctions include AMS Fellow, IEEE Fellow, SIAM Fellow, Von Humboldt award, and the NASA public service medal. Professor Bensoussan is a decorated Officer of Legion d’Honneur, Commandeur Ordre National du Merite from France and Officer Bundes Verdienst Kreuz from Germany. He has received the W.T. and Idalia Reid Prize from SIAM in 2014.

He has an extensive research background in stochastic control, risk analysis and decision making. He has published 13 books and more than 400 papers and proceedings. He develops a comprehensive approach to Risk Analysis, to apprehend technical and socio-economic risks simultaneously. He has experience in aerospace and information technology industries. His main focus is presently in the energy sector.

**[5] Speaker: Prof. Youssef Marzouk, MIT, November 2, 2022**

**Title: **

**Abstract. **

**Bio-sketch**: