Knowledge graphs represent human knowledge as a directed graph, representing each fact as a triplet consisting of a head entity, a relation, and a tail entity. Knowledge graph embedding is a representation learning technique that aims to convert the entities and relations into a set of low-dimensional embedding vectors while preserving the inherent structure of the given knowledge graph. Once the entities and relations in a knowledge graph are represented as a set of feature vectors, those vectors can be easily integrated into diverse downstream tasks. This talk introduces a new concept of knowledge graph called a bi-level knowledge graph, where the higher-level relationships between triplets can be represented. Learning representations on a bi-level knowledge graph, machines are allowed to solve problems requiring more advanced reasoning than simple link prediction. Also, as a practical example of knowledge graph embedding, how one can utilize the knowledge representations to operate a real robot is briefly explained. This talk discusses how knowledge graph embedding models effectively deliver human knowledge to machines, which is critical in many AI applications.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701 ACMseminar mailing list registration: https://mathsci.kaist.ac.kr/mailman/listinfo/acmseminar

In this talk, we provide an overview of the historical development of fast solution methods for partial differential equations, as well as their current status and potential for future advancements. We first begin with a historical survey and describe recent advances in efficient techniques, such as multigrid and domain decomposition methods. In addition, we will explore the potential of emerging methods in the realm of scientific machine learning.

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

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.

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.

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.