Speaker : Dr. Sehun Chun (Yonsei University)

Title: Moving frames for scientific computing and beyond in applications to Meteorology, Cardiology, and Neuroscience.

Abstract: First introduced as orthonormal basis vectors in the numerical solution of PDEs on curved surfaces, moving frame algorithms have been proved competitively accurate and stable for various PDEs, particularly with high-order discretization schemes. The PDEs include conservational laws, diffusion equations, shallow water equations, and Maxwell’s equations. High-order discretization schemes mean continuous/discontinuous Galerkin method or spectral/hp methods. The most striking feature of moving frames is that moving frames simplify the type of medium in PDEs. A simple representation of anisotropy by the adjusted length of the frames in diffusion equations or a general representation of rotation surfaces by moving frames provides significant advantages in numerical algorithms. Beyond the spatial representation of the complex domain, moving frames aligned along with wave propagation yields connection and Riemann curvature tensor to help to identify and predict the flow pattern. One application of such an algorithm is to analyze the cardiac electric flow where a large amount of a specific component of the Riemann curvature tensor implies conduction block and consequently the possibility of reentry and fibrillation. Another application is to construct a numerical algorithm to simulate neural spike propagation along with a spreading neural fiber bundle in the brain’s white matter to reveal the geometric structure of the brain connectivity. The most recent research also applies moving frames to a field of ‘time’ to achieve the spacetime analysis of time-dependent propagation in the heart and brain. All these moving frames applications demonstrate the beautiful simplicity of moving frames in the complex propagation phenomena in complex domains. However, a question still hangs in the air about its restrictions and real-world interpretation of connection and curvature.

Speaker : Keunsu Kim (POSTECH)

Title : Miner for a heart of gold

Abstract : 제가 하고 있는 연구 방향은 크게 두 가지입니다. 1. Time series data(1-parameter data)에 Persistent homology 이론을 적용하여 topological feature를 vectorization 하는 것. 2. Nonnegative Matrix Factorization과 Topological Data Analysis를 결합하는 방법에 대해 간략하게 소개를 하려고 합니다.

시간이 허락한다면 개인홈페이지 (https://sites.google.com/view/keunsukim/)를 바탕으로 저를 소개를 하려고 합니다.

Abstract : My direction of research has two primary reasons. 1. Apply Topological data analysis to Time series data, extract topological feature and convert to vector for using Machine Learning technic. 2. Combine Nonnegative Matrix Factorization and Topological data analysis.

If time permit, I will introduce myself based on my website. (https://sites.google.com/view/keunsukim/)

Speaker : Jeahan Jung (POSTECH)

Title : Bayesian neural networks for quantifying uncertainty in stochastic partial differential equations.

Abstract : Solving stochastic partial differential equations (SPDEs) is a challenging task in uncertainty quantification. We propose a new and novel deep learning method to solve SPDEs using Bayesian neural networks (BNNs). BNNs provide a natural approach to modeling the solutions to SPDEs since BNNs learn the distribution of their parameters so that the output of the BNN follows the target distributions. Our framework is applicable to both forward and inverse problems of SPDEs. Numerical examples are presented to illustrate the efficacy of the proposed method.

Speaker : SungWoong Cho (POSTECH)

Title : Traveling wave solutions of PDEs via neural networks]{Traveling wave solutions of partial differential equations via neural networks

Abstract : We focuses on how to approximate traveling wave solutions for various kinds of partial differential equations via artificial neural networks. A traveling wave solution is hard to obtain with traditional numerical methods when the corresponding wave speed is unknown in advance. We propose a novel method to approximate both the traveling wave solution and the unknown wave speed via a neural network and an additional free parameter. We proved that under a mild assumption, the neural network solution converges to the analytic solution and the free parameter accurately approximates the wave speed as the corresponding loss tends to zero for the Keller--Segel equation. We also demonstrate in the experiments that reducing loss through training assures an accurate approximation of the traveling wave solution and the wave speed for the Keller--Segel equation, the Allen--Cahn model with relaxation, and the Lotka--Volterra competition model.

Speaker : Taeyeob Lee (POSTECH)

Title : Interior jump and regularity for the Naiver-Stokes equations in a nonsmooth domain

Abstract : The elliptic boundary value problems have been studied in smooth domains. However, the general issues in practice are posed in domains which are simple but not smooth. So the study of the problems in nonsmooth domains is essential. We investigate the simplified stationary compressible Navier-Stokes equations in a cut domain, which is a nonsmooth boundary. We prove piecewise regularity by splitting the corner singularity functions at the cut tip and constructing the vector field which lifts the pressure jump value on the streamline.

Speaker : Jae Yong Lee (POSTECH)

Title : The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach

Abstarct : The model reduction of the mesoscopic kinetic dynamics to the macroscopic continuum dynamics has been one of the main questions in the studies of mathematical physics since Hilbert's time in the early 20th century. In this talk, we consider a diagram of diffusion limit from the Vlasov-Poisson-Fokker-Planck (VPFP) system on a bounded interval with the specular reflection boundary condition to the Poisson-Nernst–Planck (PNP) system with the no-flux boundary condition. We provide a Deep Learning algorithm to simulate the VPFP system and the PNP system via computing the time-asymptotic behaviors of the solution and the physical quantities. We analyze the convergence of the neural network solution of the VPFP system to that of the PNP system in the sense of the Asymptotic-Preserving (AP) scheme. Also, we provide theoretical evidence that Deep Neural Network (DNN) solutions to the VPFP and the PNP systems converge to the a priori classical solution of each system if the total loss function vanishes.