Special Session 57: Dynamics and Numerics of Stochastic Differential Equations

On the random age structured model

Jiaqi Cheng Northeast Normal University Peoples Rep of China Co-Author(s):    Xiaoying Han

Abstract: A random age-structured model with nonlinear birth is formulated. Its mathematical theories including wellposedness, cocycle property, and the asymptotic behaviors of the solution are developed. The emphasis is given to the asymptotic smoothness and the bounded dissipativeness of the cocycle, which implies the fractal dimension of the random attractor is finite.

Wasserstein Hamiltonian Flow and Its Structure Preserving Numerical Scheme

Jianbo Cui Hong Kong Polytechnic University Hong Kong Co-Author(s):    Luca Dieci and Haomin Zhou

Abstract: We study discretizations of Hamiltonian systems on the probability density manifold equipped with the L2-Wasserstein metric. For low dimensional problems, based on discrete optimal transport theory, several Wasserstein Hamiltonian flows (WHFs) on graph are derived. They can be viewed as spatial discretizations to the original systems. By regularizing the system using Fisher information, we propose a novel regularized symplectic scheme which could preserve several desirable longtime behaviors. Furthermore, we use the coupling idea and WHF to propose a supervised learning scheme for some high-dimensional problem.

Exponential Decay for a Klein-Gordon-Schr\{o}dinger System with locally Distributed Damping

michail E Filippakis University of Piraeus, Department of Digital Systems Greece Co-Author(s):    Poulou M.

Abstract: \begin{document} \footnote{The publication of this paper has been partly supported by the University of Pireaus Research Center}} \begin{abstract} \noindent The following coupled damped Klein-Gordon-Schr\{o}dinger equations are considered \begin{eqnarray*} i\psi_{t} +\kappa \Delta \psi+i\alpha b(x)\psi &=& \phi \psi \chi(\omega),\ \phi_{tt}- \Delta \phi +\phi +\lambda(x) \phi_t &=&- Re \grad \psi \chi(\omega), \end{eqnarray*} \noindent where $\Omega$ is a bounded domain of $\mathbb{R}^{2},$ with smooth boundary $\Gamma$ and $\omega$ is a neighbourhood of $\partial \Omega$ satisfying the geometric control condition. The aim of the paper is to prove the existence, uniqueness and uniform decay for the solutions. \end{document} \end{abstract}

A Smoluchowski-Kramers approximation to the variational wave equation

Billel Guelmame ENS Lyon France Co-Author(s):    Julien Vovelle

Abstract: We study the variational wave equation subject to stochastic forcing, which arises in the modeling of liquid crystals. In this talk, we focus on the existence of local-in-time regular solutions, the occurrence of finite-time blow-up, and the existence of global martingale weak solutions. Additionally, we explore the small-mass limit, known as the Smoluchowski-Kramers approximation, proving that the solution converges to that of a stochastic quasilinear parabolic equation. This is a joint work with Julien Vovelle.

Can one hear the shape of high-dimensional landscape?

Shirou Wang Jilin University Peoples Rep of China Co-Author(s):    Yao Li, Molei Tao

Abstract: Potential functions used in optimizations, dynamics applications, and machine learning etc. can be rather complicated in term of their structures and properties especially in very high dimensions. Due to lacking of knowledge on concrete forms of potential functions in real applications, even the determination of their basic structures and properties is a challenging problem in both mathematical analysis and numerical simulations. This talk presents a probabilistic approach to investigate the landscape of potential functions, including those in high dimensions, by using an appropriate coupling scheme to couple two copies of the overdamped Langevin dynamics of the potential functions. It can be theoretically shown that for potential functions with single or multiple wells, the coupling time distributions admit qualitatively distinct exponential tails in terms of noise magnitudes. In addition, a quantitative characterization of the non-convexity of a multi-well potential function can also be obtained via linear extrapolation. These theoretical findings thus suggest a promising approach to probe the shape of a potential landscape through the coupling time distributions at least numerically. Numerical examples of loss landscapes of neural networks with different sizes will be presented. This talk is mainly based on a recent joint work with Yao Li at UMASS and Molei Tao at Georgia Tech.

Pullback measure random attractors of lattice FitzHugh-Nagumo systems

Weisong Zhou Chongqing University of Posts and Telecommunications Peoples Rep of China Co-Author(s):    Li Dingshi, Wang Xiaohu

Abstract: In this talk, we will investigate a class of the existence of the random attractor of stochastic equations in an infinite lattice with multiplicative white noise. Using the transform, we firstly show the existence of an absorbing set, then prove that the random dynamical system is asymptotically compact. Finally, the existence of the random attractor is provided.

Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems

Zimu Zhu Hong Kong University of Science and Technology(Guangzhou) Peoples Rep of China Co-Author(s):    Chengfan Gao, Siping Gao, Ruimeng Hu

Abstract: The optimal stopping problem is one of the core problems in financial markets, with broad applications such as pricing American and Bermudan options. The deep BSDE method [Han, Jentzen and E, PNAS, 115(34):8505-8510, 2018] has shown great power in solving high-dimensional forward-backward stochastic differential equations (FBSDEs), and inspired many applications. However, the method solves backward stochastic differential equations (BSDEs) in a forward manner, which can not be used for optimal stopping problems that in general require running BSDE backwardly. To overcome this difficulty, a recent paper [Wang, Chen, Sudjianto, Liu and Shen, arXiv:1807.06622, 2018] proposed the backward deep BSDE method to solve the optimal stopping problem. In this paper, we provide the rigorous theory for the backward deep BSDE method. Specifically, 1. We derive the {\it a posteriori} error estimation, i.e., the error of the numerical solution can be bounded by the training loss function; and; 2. We give an upper bound of the loss function, which can be sufficiently small subject to universal approximations. We give two numerical examples, which present consistent performance with the proved theory. This is a joint work with C.Gao, S.Gao and R.Hu.