Deep learning techniques are increasingly applied to scientific problems, where the precision of networks is crucial. Despite being deemed as universal function approximators, neural networks, in practice, struggle to reduce the prediction errors below O(10−5) even with large network size and extended training iterations. To address this issue, we developed the multi-stage neural networks that divides the training process into different stages, with each stage using a new network that is optimized to fit the residue from the previous stage. Across successive stages, the residue magnitudes decreases substantially and follows an inverse power-law relationship with the residue frequencies. The multi-stage neural networks effectively mitigate the spectral biases associated with regular neural networks, enabling them to capture the high frequency feature of target functions. We demonstrate that the prediction error from the multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine-precision O(10−16) of double-floating point within a finite number of iterations. Such levels of accuracy are rarely attainable using single neural networks alone.

This is an introductory reading seminar presented by a senior undergraduate student, Jaehak Lee, who is studying the subject.

Link prediction (LP), inferring the connectivity between nodes, is a significant research area in graph data, where a link represents essential information on relationships between nodes. Although graph neural network (GNN)-based models have achieved high performance in LP, understanding why they perform well is challenging because most comprise complex neural networks. We employ persistent homology (PH), a topological data analysis method that helps analyze the topological information of graphs, to explain the reasons for the high performance. We propose a novel method that employs PH for LP (PHLP) focusing on how the presence or absence of target links influences the overall topology. The PHLP utilizes the angle hop subgraph and new node labeling called degree double radius node labeling (Degree DRNL), distinguishing the information of graphs better than DRNL. Using only a classifier, PHLP performs similarly to state-of-the-art (SOTA) models on most benchmark datasets. Incorporating the outputs calculated using PHLP into the existing GNN-based SOTA models improves performance across all benchmark datasets. To the best of our knowledge, PHLP is the first method of applying PH to LP without GNNs. The proposed approach, employing PH while not relying on neural networks, enables the identification of crucial factors for improving performance. https://arxiv.org/abs/2404.15225

최신 논문 리뷰: Rapid Convergence of Unadjusted Langevin Algorithm (Vempala et al) and Score-Based Generative Models(Song et al)

I tell a personal story of how a mathematician working in complex algebraic geometry had come to discover the relevance of Cartan geometry, a subject in differential geometry, in an old problem in algebraic geometry, the problem of deformations of Grassmannians as projective manifolds, which originated from the work of Kodaira and Spencer. In my joint work with Ngaiming Mok, we used the theory of minimal rational curves to study such deformations and it reduced the question to a problem in Cartan geometry.

We introduce a general equivalence problems for geometric structures arising from minimal rational curves on uniruled complex projective manifolds. To study these problems, we need approaches fusing differential geometry and algebraic geometry. Among such geometric structures, those associated to homogeneous manifolds are particularly accessible to differential-geometric methods of Cartan geometry. But even in these cases, only a few cases have been worked out so far. We review some recent developments.

This lecture explores the topics and areas that have guided my research in computational mathematics and machine learning in recent years. Numerical methods in computational science are essential for comprehending real-world phenomena, and deep neural networks have achieved state-of-the-art results in a range of fields. The rapid expansion and outstanding success of deep learning and scientific computing have led to their applications across multiple disciplines, ranging from fluid dynamics to material sciences. In this lecture, I will focus on bridging machine learning with applied mathematics, specifically discussing topics such as scientific machine learning, numerical PDEs, and mathematical approaches of machine learning, including generative models and adversarial examples.

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.

1. 데이터 분석 업무의 이해(김준범)- 데이터 분석가의 역할 소개 2. 초거대 언어 모델 동향(김정섭)-GPT-3 부터 Llama-3까지 이미 우리 삶 속에 깊숙이 자리잡은 초거대 언어 모델의 동향 3. 데이터 분석가에서 공직으로 오게된 과정과 앞으로의 계획(심규석)- 삼성화재에서의 데이터 분석 및 AI 모델링 업무, 행정안전부에서의 데이터 분석과제 기획·관리 및 공무원의 데이터 분석 역량지원 업무 전반에 관한 설명과 함께 각 기관을 지원하게 된 동기, 지원방법, 준비사항 등

This is an introductory reading seminar presented by a senior undergraduate student, Jaehak Lee, who is studying the subject.

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.

The finite quotient groups of étale fundamental groups of algebraic curves in positive characteristic are precisely determined, but without explicit construction of quotient maps, by well-known results of Raynaud, Harbater and Pop, previously known as Abhyankar's conjecture. Katz, Rojas León and Tiep have been studying the constructive side of this problem using certain "easy to remember" local systems. In this talk, I will discuss the main results and methods of this project in the case of a specific type of local systems called hypergeometric sheaves.

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In this talk, we consider the Boltzmann equation in general 3D toroidal domains with a specular reflection boundary condition. So far, it is a well-known open problem to obtain the low-regularity solution for the Boltzmann equation in general non-convex domains because there are grazing cases, such as inflection grazing. Thus, it is important to analyze trajectories which cause grazing. We will provide new analysis to handle these trajectories in general 3D toroidal domains.

박사후 연구 이야기- 박사과정 이후의 삶을 대비하는 방법, 국내 및 해외에서의 박사후연구원 생활과 필요한 준비사항 등