While deep neural networks (DNNs) have been widely used in numerous applications over the past few decades, their underlying theoretical mechanisms remain incompletely understood. In this presentation, we propose a geometrical and topological approach to understand how deep ReLU networks work on classification tasks. Specifically, we provide lower and upper bounds of neural network widths based on the geometrical and topological features of the given data manifold. We also prove that irrespective of whether the mean square error (MSE) loss or binary cross entropy (BCE) loss is employed, the loss landscape has no local minimum.

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

Dirac's theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering hypergraph matchings and Hamiltonian cycles. We consider another natural generalization of the perfect matchings, Steiner triple systems. As a Steiner triple system can be viewed as a partition of pairs of vertices, it is a natural high-dimensional analogue of a perfect matching in graphs. We prove that for sufficiently large integer $n$ with $n \equiv 1 \text{ or } 3 \pmod{6},$ any $n$-vertex $3$-uniform hypergraph $H$ with minimum codegree at least $\left(\frac{3 + \sqrt{57}}{12} + o(1) \right)n = (0.879... + o(1))n$ contains a Steiner triple system. In fact, we prove a stronger statement by considering transversal Steiner triple systems in a collection of hypergraphs. We conjecture that the number $\frac{3 + \sqrt{57}}{12}$ can be replaced with $\frac{3}{4}$ which would provide an asymptotically tight high-dimensional generalization of Dirac's theorem.

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심사위원장: 이지운, 심사위원: 남경식, 황강욱, 양홍석(전산학부), 폴정(Fordham University)

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

In the analysis of singularities, uniqueness of limits often arises as an important question: that is, whether the geometry depends on the scales one takes to approach the singularity. In his seminal work, Simon demonstrated that Lojasiewicz inequalities, originally known in real algebraic geometry in finite dimensions, can be applied to show uniqueness of limits in geometric analysis in infinite dimensional settings. We will discuss some instances of this very successful technique and its applications.

Finite path integral is a finite version of Feynman’s path integral, which is a mathematical methodology to construct TQFT’s (topological quantum field theories) from finite gauge theory. It was introudced by Dijkgraaf and Witten in 1990. We study finite path integral model by replacing finite gauge theory with homological algebra based on bicommutative Hopf algebras. It turns out that Mayer-Vietoris functors such as homology theories extend to TQFT which preserves compositions up to a scalar. This talk concerns the second cohomology class of cobordism (more generally, cospan) categories induced by such scalars. In particular, we will explain that the obstruction class is described purely by homological algebra, not via finite path integral.

I will report on some recent results on modelling the heart, the external circulation, and their application to problems of clinical relevance. I will show that a proper integration between PDE-based and machine-learning algorithms can improve the computational efficiency and enhance the generality of our iHEART simulator.

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234) + Google Map

Zeta functions and zeta values play a central role in Modern Number Theory and are connected to practical applications in codes and cryptography. The significance of these objects is demonstrated by the fact that two of the seven Clay Mathematics Million Dollar Millennium Problems are related to these objects, namely the Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture. We first recall results and well-known conjectures concerning these objects over number fields. If time permits, we will present recent developments in the setting of function fields. This is a joint work with Im Bo-Hae and Kim Hojin among others.

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There will be a tea time at 15:30 before the lecture.
Contact: Professor Bo-Hae Im ()

https://mathsci.kaist.ac.kr/bk21four/index.php/boards/view/board_seminar/3/

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심사위원장: 김용정, 심사위원: 권순식, 강문진, 김재경, 윤창욱(충남대학교)

The Stefan problem is a free boundary problem describing the interface between water and ice. It has PDE and probabilistic aspects. We discuss an approach to this problem, based on optimal transport theory. This approach is related to the Skorokhod problem, a classical problem in probability regarding the Brownian motion.

The mapping class group Map(S) of a surface S is the group of isotopy classes of diffeomorphisms of S. When S is a finite-type surface, the classical mapping class group Map(S) has been well understood. On the other hand, there are recent developments on mapping class groups of infinite-type surfaces. In this talk, we discuss mapping class groups of finite-type and infinite-type surfaces and elements of these groups. Also, we define surface Houghton groups, which are subgroups of mapping class groups of certain infinite-type surfaces. Then we discuss finiteness properties of surface Houghton groups, which is a joint work with Aramayona, Bux, and Leininger.

For a uniformly supersonic flow past a convex cornered wedge with the pressure being given for the surrounding quiescent gas at the downstream, as shown in experimental results, it is expected to form a shock followed by a contact discontinuity, which is also called the jet flow. By the shock polar analysis, it is well-known that there are two possible shocks, one a strong shock and the other one a weak shock. The strong shock is always transonic, while the weak shock could be transonic or supersonic. We prove the global existence, asymptotic behaviors, uniqueness, and stability of the subsonic jet with a strong transonic shock under the perturbation of the upstream flow and the pressure of the surrounding quiescent gas, for the two-dimensional steady full Euler equations.

Given a hypergraph $H=(V,E)$, we say that $H$ is (weakly) $m$-colorable if there is a coloring $c:V\to [m]$ such that every hyperedge of $H$ is not monochromatic. The (weak) chromatic number of $H$, denoted by $\chi(H)$, is the smallest $m$ such that $H$ is $m$-colorable. A vertex subset $T \subseteq V$ is called a transversal of $H$ if for every hyperedge $e$ of $H$ we have $T\cap e \ne \emptyset$. The transversal number of $H$, denoted by $\tau(H)$, is the smallest size of a transversal in $H$. The transversal ratio of $H$ is the quantity $\tau(H)/|V|$ which is between 0 and 1. Since a lower bound on the transversal ratio of $H$ gives a lower bound on $\chi(H)$, these two quantities are closely related to each other. Upon my previous presentation, which is based on the joint work with Joseph Briggs and Michael Gene Dobbins (https://www.youtube.com/watch?v=WLY-8smtlGQ), we update what is discovered in the meantime about transversals and colororings of geometric hypergraphs. In particular, we focus on chromatic numbers of $k$-uniform hypergraphs which are embeddable in $\mathbb{R}^d$ by varying $k$, $d$, and the notion of embeddability and present lower bound constructions. This result can also be regarded as an improvement upon the research program initiated by Heise, Panagiotou, Pikhurko, and Taraz, and the program by Lutz and Möller. We also present how this result is related to the previous results and open problems regarding transversal ratios. This presentation is based on the joint work with Eran Nevo.

We consider a family of nonlocal diffusion equations with a prescribed equilibrium state, which includes the fractional heat equation as well as a nonlocal equation of Fokker-Planck type. This family of equations will be shown to arise as the gradient flow of the relative entropy with respect to a version of the nonlocal Wasserstein metric introduced by Erbar. Such equations may also be viewed as the evolutionary Gamma-limit of a certain sequence of heat flows on discrete Markov chains. I will discuss criteria for existence, uniqueness, and stability of solutions, and sufficient criteria on the equilibrium state which ensure fast convergence to equilibrium.

TBD

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234) + Google Map

In this talk, we consider a group-sparse matrix estimation problem. This problem can be solved by applying the existing compressed sensing techniques, which either suffer from high computational complexities or lack of algorithm robustness. To overcome the situation, we propose a novel algorithm unrolling framework based on the deep neural network to simultaneously achieve low computational complexity and high robustness. Specifically, we map the original iterative shrinkage thresholding algorithm (ISTA) into an unrolled recurrent neural network (RNN), thereby improving the convergence rate and computational efficiency through end-to-end training. Moreover, the proposed algorithm unrolling approach inherits the structure and domain knowledge of the ISTA, thereby maintaining the algorithm robustness, which can handle non-Gaussian preamble sequence matrix in massive access. We further simplify the unrolled network structure with rigorous theoretical analysis by reducing the redundant training parameters. Furthermore, we prove that the simplified unrolled deep neural network structures enjoy a linear convergence rate. Extensive simulations based on various preamble signatures show that the proposed unrolled networks outperform the existing methods regarding convergence rate, robustness, and estimation accuracy.

In this talk, we consider nonlinear elliptic equations of the $p$-Laplacian type with lower order terms which involve nonnegative potentials satisfying a reverse H\"older type condition. We establish interior and boundary $L^q$ estimates for the gradient of weak solutions and the lower order terms, independently, under sharp regularity conditions on the coefficients and the boundaries. In addition, we prove interior estimates for Hessian of strong solutions and the lower order terms for nondivergence type elliptic equations. The talk is based on joint works with Jihoon Ok and Yoonjung Lee.

In this talk, we will primarily discuss the theoretical analysis of knowledge distillation based federated learning algorithms. Before we explore the main topics, we will introduce the basic concepts of federated learning and knowledge distillation. Subsequently, we will understand a nonparametric view of knowledge distillation based federated learning algorithms and introduce generalization analysis of these algorithms based the theory of regularized kernel regression methods.

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

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심사위원장: 백상훈, 심사위원: 곽시종, 김완수, 이용남, 쾨니히 요아힘(한국교원대학교)

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심사위원장: 임보해, 심사위원: 김완수, 박진형, 쾨니히 요아힘(한국교원대학교), 조재현(UNIST)

Interpreting data using mechanistic mathematical models provides a foundation for discovery and decision-making in all areas of science and engineering. Key steps in using mechanistic mathematical models to interpret data include: (i) identifiability analysis; (ii) parameter estimation; and (iii) model prediction. Here we present a systematic, computationally efficient likelihood-based workflow that addresses all three steps in a unified way. Recently developed methods for constructing profile-wise prediction intervals enable this workflow and provide the central linkage between different workflow components. These methods propagate profile-likelihood-based confidence sets for model parameters to predictions in a way that isolates how different parameter combinations affect model predictions. We show how to extend these profile-wise prediction intervals to two-dimensional interest parameters, and then combine profile-wise prediction confidence sets to give an overall prediction confidence set that approximates the full likelihood-based prediction confidence set well. We apply our methods to a range of synthetic data and real-world ecological data describing re-growth of coral reefs on the Great Barrier Reef after some external disturbance, such as a tropical cyclone or coral bleaching event.

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234) + Google Map

In this talk, I will introduce the use of deep neural networks (DNNs) to solve high-dimensional evolution equations. Unlike some existing methods (e.g., least squares method/physics-informed neural networks) that simultaneously deal with time and space variables, we propose a deep adaptive basis approximation structure. On the one hand, orthogonal polynomials are employed to form the temporal basis to achieve high accuracy in time. On the other hand, DNNs are employed to create the adaptive spatial basis for high dimensions in space. Numerical examples, including high-dimensional linear parabolic and hyperbolic equations and a nonlinear Allen–Cahn equation, are presented to demonstrate that the performance of the proposed DABG method is better than that of existing DNNs. zoom link: https://kaist.zoom.us/j/3844475577 zoom ID: 384 447 5577

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https://kaist.zoom.us/j/3844475577 회의 ID: 384 447 5577

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

In this talk, we address a question whether a mean-field approach for a large particle system is always a good approximation for a large particle system or not. For definiteness, we consider an infinite Kuramoto model for a countably infinite set of Kuramoto oscillators and study its emergent dynamics for two classes of network topologies. For a class of symmetric and row (or columm)-summable network topology, we show that a homogeneous ensemble exhibits complete synchronization, and the infinite Kuramoto model can cast as a gradient flow, whereas we obtain a weak synchronization estimate, namely practical synchronization for a heterogeneous ensemble. Unlike with the finite Kuramoto model, phase diameter can be constant for some class of network topologies which is a novel feature of the infinite model. We also consider a second class of network topology (so-called a sender network) in which coupling strengths are proportional to a constant that depends only on sender's index number. For this network topology, we have a better control on emergent dynamics. For a homogeneous ensemble, there are only two possible asymptotic states, complete phase synchrony or bi-cluster configuration in any positive coupling strengths. In contrast, for a heterogeneous ensemble, complete synchronization occurs exponentially fast for a class of initial configuration confined in a quarter arc. This is a joint work with Euntaek Lee (SNU) and Woojoo Shim (Kyungpook National University).

The spectrum of a general non-Hermitian (non-normal) matrix is unstable; a tiny perturbation of the matrix may result in a huge difference in its eigenvalues. This instability is often quantified as eigenvalue condition numbers in numerical linear algebra or as eigenvector overlap in random matrix theory. In this talk, we show that adding a smoothly random noise matrix regularizes this instability, by proving a nearly optimal upper bound of eigenvalue condition numbers. If time permits, we will also discuss the effect of the noise matrix on a macroscopic scale in terms of the Brown measure of free circular Brownian motion. This talk is based on joint works with László Erdős.

Determining the density required to ensure that a host graph G contains some target graph as a subgraph or minor is a natural and well-studied question in extremal combinatorics. The celebrated 50-year-old Erdős-Sós conjecture states that for every k, if G has average degree exceeding k-2 then it contains every tree T with k vertices as a subgraph. This is tight as the clique with k-1 vertices contains no tree with k vertices as a subgraph. We present some variants of this conjecture. We first consider replacing bounds on the average degree by bounds on the minimum and maximum degrees. We then consider replacing subgraph by minor in the statement.

In this talk, I will introduce twistor theory, which connects complex geometry, Riemannian geometry, and algebraic geometry by producing a complex manifold, called the twistor space, from a quaternionic Kähler manifold. First, I will explain why quaternionic Kähler manifolds have to be studied in view of holonomy theory in Riemannian geometry, and how twistor theory enables us to use algebraic geometry in studying their geometry. Next, based on the realization of homogeneous twistor spaces as adjoint varieties, I will present a description of the compactified spaces of conics in adjoint varieties, which is motivated by twistor theory.

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심사위원장:이지운, 심사위원:남경식, 황강욱, 윤세영(김재철AI대학원), 서성미(충남대학교)

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

(KAI-X Distinguished Lecture Series) We have multiple approaches to vanishing theorems for the cohomology of Shimura varieties, via either algebraic geometry or automorphic forms. Such theorems have been of interest with either complex or torsion coefficients. Recently, results have been obtained under various genericity hypotheses by Caraiani-Scholze, Koshikawa, Hamann-Lee et al. I will survey different approaches. If time permits, I may discuss an ongoing project with Koshikawa to understand the non-generic case.

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KAI-X Distinguished lecture

The Hypothalamic-Pituitary-Adrenal (HPA) axis is the key regulatory pathway responsible for maintaining homeostasis under conditions of real or perceived stress. Endocrine responses to stressors are mediated by adrenocorticotrophic hormone (ACTH) and corticosteroid (CORT) hormones. In healthy, non-stressed conditions, ACTH and CORT exhibit highly correlated ultradian pulsatility with an amplitude modulated by circadian processes. Disruption of these hormonal rhythms can occur as a result of stressors or in the very early stages of disease. Despite the fact that misaligned endocrine rhythms are associated with increased morbidity, a quantitative understanding of their mechanistic origin and pathogenicity is missing. Mathematically, the HPA axis can be understood as a dynamical system that is optimised to respond and adapt to perturbations. Normally, the body copes well with minor disruptions, but finds it difficult to withstand severe, repeated or long-lasting perturbations. Whilst a healthy HPA axis maintains a certain degree of robustness to stressors, its fragility in diseased states is largely unknown, and this understanding constitutes a critical step toward the development of digital tools to support clinical decision-making. This talk will explore how these challenges are being addressed by combining high-resolution biosampling techniques with mathematical and computational analysis methods. This interdisciplinary approach is helping us quantify the inter-individual variability of daily hormone profiles and develop novel “dynamic biomarkers” that serve as a normative reference and to signal endocrine dysfunction. By shifting from a qualitative to a quantitative description of the HPA axis, these insights bring us a step closer to personalised clinical interventions for which timing is key.

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234) + Google Map

We consider the problem of graph matching, or learning vertex correspondence, between two correlated stochastic block models (SBMs). The graph matching problem arises in various fields, including computer vision, natural language processing and bioinformatics, and in particular, matching graphs with inherent community structure has significance related to de-anonymization of correlated social networks. Compared to the correlated Erdos-Renyi (ER) model, where various efficient algorithms have been developed, among which a few algorithms have been proven to achieve the exact matching with constant edge correlation, no low-order polynomial algorithm has been known to achieve exact matching for the correlated SBMs with constant correlation. In this work, we propose an efficient algorithm for matching graphs with community structure, based on the comparison between partition trees rooted from each vertex, by extending the idea of Mao et al. (2021) to graphs with communities. The partition tree divides the large neighborhoods of each vertex into disjoint subsets using their edge statistics to different communities. Our algorithm is the first low-order polynomial-time algorithm achieving exact matching between two correlated SBMs with high probability in dense graphs.

We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd integral distance from each other.

: For a translation surface, the associated saddle connection graph has saddle connections as vertices, and edges connecting pairs of non-crossing saddle connections. This can be viewed as an induced subgraph of the arc graph of the surface. In this talk, I will discuss both the fine and coarse geometry of the saddle connection graph. We show that the isometry type is rigid: any isomorphism between two such graphs is induced by an affine diffeomorphism between the underlying translation surfaces. However, the situation is completely different when one considers the quasi-isometry type: all saddle connection graphs form a single quasi-isometry class. We will also discuss the Gromov boundary in terms of foliations. This is based on joint work with Valentina Disarlo, Huiping Pan, and Anja Randecker.

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심사위원장: 김동환, 심사위원: 이창옥, 홍영준, 곽도영, 오덕순(충남대학교)

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심사위원장: 김동환, 심사위원: 이창옥, 홍영준, 곽도영, 오덕순(충남대학교)

The talk will focus on a part of the frontier of our current understanding of nonlinear stability of traveling waves of partial differential equations, especially on how spectral stability implies nonlinear stability and which kind of dynamics may be expected. We shall highlight main expected difficulties related to the stability of discontinuous waves of hyperbolic systems, and show a few significant steps obtained by the speaker with respectively Vincent Duchêne (Rennes), Gregory Faye (Toulouse) and Louis Garénaux (Karslruhe).

The Gauss-Bonnet theorem implies that the two dimensional torus does not have nonnegative Gauss curvature unless it is flat, and that the two dimensional sphere does not a metric which has Gaussian curvature bounded below by one and metric bounded below by the standard round metric. Gromov proposed a series of conjectures on generalizing the Gauss-Bonnet theorem in his four lectures. I will report my work with Gaoming Wang (now Tsinghua) on Gromov dihedral rigidity conjecture in hyperbolic 3-space and scalar curvature comparison of rotationally symmetric convex bodies with some simple singularities.

In this lecture, we aim to delve deep into the emerging landscape of 'Foundation Models'. Distinct from traditional deep learning models, Foundation Models have ushered in a new paradigm, characterized by their vast scale, versatility, and transformative potential. We will uncover the key differences between these models and their predecessors, delving into the intricate mechanisms through which they are trained and the profound impact they are manifesting across various sectors. Furthermore, the talk will shed light on the invaluable role of mathematics in understanding, optimizing, and innovating upon these models. We will explore the symbiotic relationship between Foundation Models and mathematical principles, elucidating how the latter not only underpins their functioning but also paves the way for future advancements.

Graph product structure theory describes complex graphs in terms of products of simpler graphs. In this talk, I will introduce this subject and talk about some of my recent results in this area. The focus of my talk will be on a new tool in graph product structure theory called `blocking partitions.’ I’ll show how this tool can be used to prove stronger product structure theorems for powers of planar graphs as well as k-planar graphs, resolving open problems of Dujmović, Morin and Wood, and Ossona de Mendez.

In this talk, we study the non-cutoff Boltzmann collision kernel for the inverse power law potentials $U_s(r)=1/r^{s-1}$ for $s>2$ in dimension $d=3$. We will study the formal derivation of the non-cutoff collision kernel. Then we will prove the limit of the non-cutoff kernel to the hard-sphere kernel and check the angular singularity would vanish. We will also see precise asymptotic formulas of the singular layer near $\theta\simeq 0$ in the limit $s\to \infty$. Consequently, we will also see that solutions to the homogeneous Boltzmann equation converge to the respective solutions weakly in $L^1$ globally in time as $s\to \infty$.

Almost all biological systems possess the ability to gather environmental information and modulate their behaviors to adaptively respond to changing environments. While animals excel at sensing odors, even simple bacteria can detect faint chemicals using stochastic receptors. They then navigate towards or away from the chemical source by processing this sensed information through intracellular reaction systems. In the first half of our talk, we demonstrate that the E. coli chemotactic system is optimally structured for sensing noisy signals and controlling taxis. We utilize filtering theory and optimal control theory to theoretically derive this optimal structure and compare it to the quantitatively verified biochemical model of chemotaxis. In the latter half, we discuss the limitations of traditional information theory, filtering theory, and optimal control theory in analyzing biological systems. Notably, all biological systems, especially simpler ones, have constrained computational resources like memory size and energy, which influence optimal behaviors. Conventional theories don’t directly address these resource constraints, likely because they emerged during a period when computational resources were continually expanding. To address this gap, we introduce the “memory-limited partially observable optimal control,” a new theoretical framework developed by our group, and explore its relevance to biological problems.

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234) + Google Map

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심사위원장: 변재형, 심사위원 : 강문진, 김용정, 배명진, 권오상(충북대학교)

An edge-coloured graph is said to be rainbow if it uses no colour more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on n vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of $(\log n)^{2+o(1)}$ for this question. Very recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the $o(1)$ term in Tomon's bound. We show that the answer to the question is equal to $(\log n)^{1+o(1)}$. A key tool we use is the theory of robust sublinear expanders. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non-abelian groups. Joint work with: Noga Alon, Lisa Sauermann, Dmitrii Zakharov and Or Zamir.

With the success of deep learning technologies in many scientific and engineering applications, neural network approximation methods have emerged as an active research area in numerical partial differential equations. However, the new approximation methods still need further validations on their accuracy, stability, and efficiency so as to be used as alternatives to classical approximation methods. In this talk, we first introduce the neural network approximation methods for partial differential equations, where a neural network function is introduced to approximate the PDE (Partial Differential Equation) solution and its parameters are then optimized to minimize the cost function derived from the differential equation. We then present the approximation error and the optimization error behaviors in the neural network approximate solution. To reduce the approximation error, a neural network function with a larger number of parameters is often employed but when optimizing such a larger number of parameters the optimization error usually pollutes the solution accuracy. In addition to that, the gradient-based parameter optimization usually requires computation of the cost function gradient over a tremendous number of epochs and it thus makes the cost for a neural network solution very expensive. To deal with such problems in the neural network approximation, a partitioned neural network function can be formed to approximate the PDE solution, where localized neural network functions are used to form the global neural network solution. The parameters in each local neural network function are then optimized to minimize the corresponding cost function. To enhance the parameter training efficiency further, iterative algorithms for the partitioned neural network function can be developed. We finally discuss the possibilities in this new approach as a way of enhancing the neural network solution accuracy, stability, and efficiency by utilizing classical domain decomposition algorithms and their convergence theory. Some interesting numerical results are presented to show the performance of the partitioned neural network approximation and the iteration algorithms.

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

Maximal functions of various forms have played crucial roles in harmonic analysis. Various outstanding open problems are related to Lp boundedness (estimate) of the associated maximal functions. In this talk, we discuss Lp boundedness of maximal functions given by averages over curves.

In physics, Bohr’s correspondence principle asserts that the theory of quantum mechanics can be reduced to that of classical mechanics in the limit of large quantum numbers. This rather vague statement can be formulated explicitly in various ways. In this talk, focusing on an analytic point of view, we discuss the correspondence between basic inequalities and that between measures. Then, as an application, we present the convergence from quantum to kinetic white dwarfs.

The k-color induced size-Ramsey number of a graph H is the smallest number of edges a (host) graph G can have such that for any k-coloring of its edges, there exists a monochromatic copy of H which is an induced subgraph of G. In 1995, in their seminal paper, Haxell, Kohayakawa and Łuczak showed that for cycles these numbers are linear for any constant number of colours, i.e., for some C=C(k), there is a graph with at most Cn edges whose any k-edge-coloring contains a monochromatic induced cycle of length n. The value of C comes from the use of the sparse regularity lemma and has a tower-type dependence on k. In this work, we obtain nearly optimal bounds for the required value of C. Joint work with Nemanja Draganić and Benny Sudakov.

"어떻게 하면 더 좋은 제품을 더 빠르게 개발할 수 있을까?"라는 문제는 모든 제조업이 안고 있는 숙제입니다. 최근 DX를 통해 많은 데이터들이 디지털화되고, AI의 급격한 발전을 통해 제품개발프로세스를 혁신하려는 시도가 일어나고 있습니다. 과거의 시뮬레이션 기반 설계에서 AI 기반 설계로의 패러다임 전환을 통해 제품개발 기간을 단축함과 동시에 제품의 품질을 향상시킬 수 있습니다. 본 세미나는 딥러닝을 통해 제품 설계안을 생성/탐색/예측/최적화/추천할 수 있는 생성형 AI 기반의 설계 프로세스(Deep Generative Design)를 소개하고, 모빌리티를 비롯한 제조 산업에 적용된 다양한 사례들을 소개합니다.

In this talk, we discuss the Neural Tangent Kernel. The NTK is closely related to the dynamics of the neural network during training via the Gradient Flow(or Gradient Descent). But, since the NTK is random at initialization and varies during training, it is quite delicate to understand the dynamics of the neural network. In relation to this issue, we introduce an interesting result: in the infinite-width limit, the NTK converge to a deterministic kernel at initialization and remains constant during training. We provide a brief proof of the result for the simplest case.

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9월 14일, 10월 4일, 5일 세 번에 걸친 발표.

In this talk, I will explain the setting of online convex optimization and the definition of regret and constraint violation. I then will introduce various algorithms and their theoretical guarantees under various assumptions. The connection with some topics in machine learning such as stochastic gradient descent, multi-armed bandit, and reinforcement learning will also be briefly discussed.

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

In this talk, we discuss the Neural Tangent Kernel. The NTK is closely related to the dynamics of the neural network during training via the Gradient Flow(or Gradient Descent). But, since the NTK is random at initialization and varies during training, it is quite delicate to understand the dynamics of the neural network. In relation to this issue, we introduce an interesting result: in the infinite-width limit, the NTK converge to a deterministic kernel at initialization and remains constant during training. We provide a brief proof of the result for the simplest case.

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9월 14일, 10월 4일, 5일 세 번에 걸친 발표로, 본 시간에는 주로 9월 14일 내용의 리뷰를 주로 다룸.