Bayesian Physics Informed Neural Networks (B-PINNs) have gained significant attention for inferring physical parameters and learning the forward solutions for problems based on partial differential equations. However, the overparameterized nature of neural networks poses a computational challenge for high-dimensional posterior inference. Existing inference approaches, such as particle-based or variance inference methods, are either computationally expensive for highdimensional posterior inference or provide unsatisfactory uncertainty estimates. In this paper, we present a new efficient inference algorithm for B-PINNs that uses Ensemble Kalman Inversion (EKI) for high-dimensional inference tasks. By reframing the setup of B-PINNs as a traditional Bayesian inverse problem, we can take advantage of EKI’s key features: (1) gradient-free, (2) computational complexity scales linearly with the dimension of the parameter spaces, and (3) rapid convergence with typically O(100) iterations. We demonstrate the applicability and performance of the proposed method through various types of numerical examples. We find that our proposed method can achieve inference results with informative uncertainty estimates comparable to Hamiltonian Monte Carlo (HMC)-based B-PINNs with a much reduced computational cost. These findings suggest that our proposed approach has great potential for uncertainty quantification in physics-informed machine learning for practical applications.

We will first introduce Homogeneous dynamics, especially the mixing property of flows on spaces of hyperbolic nature. We will then survey applications of homogeneous dynamics to various problems in Number theory. (Part of the talk is based on joint work with Keivan Mallahi-Karai and Jiyoung Han.)

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The date has been postponed from March 16 to March 30.

Modeling mass or heat transfer near a wall is of broad interest in various fluid flows. Specifically, in cardiovascular flows, mass transport near the vessel wall plays an important role in cardiovascular disease. However, due to very thin concentration boundary layers, accurate computational modeling is challenging. Additionally, experimental approaches have limitations in measuring near-wall flow metrics such as wall shear stress (WSS). In this talk, first, I will briefly review the complex flow physics near the wall in diseased vascular flows and introduce the concept of WSS manifolds in near-wall transport. Specifically, I will talk about stable and unstable manifolds calculated for a surface vector field. Next, I will discuss reduced-order data assimilation modeling as well as physics-informed neural network (PINN) approaches for obtaining WSS from measurement data away from the wall. Finally, I present a boundary-layer PINN (BL-PINN) approach inspired by the classical perturbation theory and asymptotic expansions to solve challenging thin boundary layer mass transport problems. BL-PINN demonstrates how classical theoretical approaches could be replicated in a deep learning framework.

In this lecture we introduce some challenging problems in the mathematical fluid mechanics. Although fluid mechanics is one of the most important physical phenomena we experience in everyday life, and has been studied for long time in history by top class mathematicians, still there are many problems which are open even at the fundamental level. We explain these problems and briefly review some of the recent progress.

Symmetric spaces from Lie theory and differential geometry are often represented by special set of structured matrices. The Cartan decomposition and its generalization of symmetric spaces and classical Lie groups recover many of the known matrix factorizations in numerical linear algebra, such as the singular value decomposition, CS decomposition, generalized SVD and many more. We discuss a blueprint for generating fifty-three matrix factorizations from the generalized Cartan decomposition, most of which appear to be new. The underlying mathematics may be traced back to Cartan (1927), Harish-Chandra (1956), and Flensted-Jensen (1978). This is joint work with Alan Edelman.

Many differential equations and partial differential equations (PDEs) are being studied to model physical phenomena in nature with mathematical expressions. Recently, new numerical approaches using machine learning and deep learning have been actively studied. There are two mainstream deep learning approaches to approximate solutions to the PDEs, i.e., using neural networks directly to parametrize the solution to the PDE and learning operators from the parameters of the PDEs to their solutions. As the first direction, Physics-Informed Neural Network was introduced in (Raissi, Perdikaris, and Karniadakis 2019), which learns the neural network parameters to minimize the PDE residuals in the least-squares sense. On the other side, operator learning using neural networks has been studied to approximate a PDE solution operator, which is nonlinear and complex in general. In this talk, I will introduce these two ways to approximate the solution of PDE and my research related to them.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701

This talk reviews two notable papers in self-supervised graphical neural networks; they are "Graph contrastive learning with augmentations" presented at NeurIPS 2020 and "Contrastive multi-view representation learning on graphs" presented at ICML 2020. This will be an introduction of self-supervised graphical neural networks that has emerged as one of the hottest research fields in artificial intelligence, which requires mathematical methodology across all fields of mathematics, including graph theory, algebra, topology, analysis, and geometry.

In this talk, we study the dissipative structure for the linear symmetric hyperbolic system with general relaxation. If the relaxation matrix of the system has symmetric properties, Shizuta and Kawashima(1985) introduced the suitable stability condition, and Umeda, Kawashima and Shizuta(1984) analyzed the dissipative structure. On the other hand, Ueda, Duan and Kawashima(2012,2018) focused on the system with non-symmetric relaxation and got partial results. Furthermore, they argued the new dissipative structure called the regularity-loss type. In this situation, this talk aims to extend the stability theory introduced by Shizuta and Kawashima(1985) and Umeda, Kawashima and Shizuta(1984) to our general system. Furthermore, we will consider the optimality of the dissipative structure. If we have time, I would like to discuss some physical models for its application and new dissipative structures.

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)