일시:  2025년 04월 02일(월) 11:00~12:00

장소:  Zoom meeting

연사: 오재민 박사 (Texas A&M University)

주제: Machine Learning Approaches to Differential Equations

내용: Numerically solving differential equations is a central focus in scientific computing and numerical analysis. Polynomial-based methods are widely used due to their strong theoretical foundation and efficient implementation, making them indispensable in computational simulations. However, these methods also present challenges, such as the curse of dimensionality and a steep learning curve. In this talk, I will explore machine learning approaches to differential equations that may help overcome these limitations, starting with physics-informed neural networks (PINNs). I will first present my recent work on the Boltzmann-BGK model and lithium-ion battery simulations. Then, I will discuss the potential benefits and limitations of PINNs. Finally, I will briefly introduce neural ordinary differential equation-based methods as alternative approaches.

일시:  2024년 11월 11일(월) 16:00~17:00

장소:  경북대학교 자연과학대학 319호

연사:  한지훈 박사 (다트머스대학)

주제: A stochastic approach for solving PDEs: derivative-free loss method

강의내용:   I will discuss the derivative-free loss method (DFLM), a stochastic approach for solving PDEs. The method uses the stochastic representation of the PDE in the spirit of the Feynman-Kac formula. It characterizes the averaging of collective information from stochastic walkers’ paths exploring the neighborhood of a point of interest. While exploring the domain with an iterative averaging process, a neural network is reinforced to approximate the PDE solution. I will cover its analysis regarding trainability and highlight its effectiveness in non-intrusively tackling multiscale problems with highly oscillating coefficients and perforated domain problems.

일시:  2024년 10월 18일(금) 15:00~16:00

장소:  경북대학교 자연과학대학 213호

연사:  정소연 교수 (공주대학교 국제학부)

주제: Traveling waves for monostable reaction-diffusion-convection equations with discontinuous density-dependent coefficients. 

강의내용:   In this talk we consider wave propagation in a class of scalar reaction-diffusion-convection equations with p-Laplacian-type diffusion and monostable reaction. We introduce a new concept of a non-smooth traveling wave profile, which allows us to treat discontinuous diffusion with possible degenerations and singularities at 0 and 1, as well as only piecewise continuous convective velocity. Our approach is based on comparison arguments for an equivalent non-Lipschitz first-order ODE. We formulate sufficient conditions for the existence and non-existence of these generalized solutions and discuss how the convective velocity affects the minimal wave speed compared to the problem without convection. 

일시:  2024년 5월 24일 (금) 15:00~15:50 / 16:00~16:50

장소:  자연과학대학 219호

연사: 강문진 교수님 (KAIST)

강의내용:  The Cauchy problem for compressible Euler system from inviscid limit of Navier-Stokes remains completely open, as a challenging issue in fluid dynamics. This lectures focus on the 1D isentropic case for the open problem. We will present the global well-posedness of entropy solutions with small BV initial data in the class of inviscid limits from the associate Navier-Stokes. More precisely, any small BV entropy solutions are inviscid limits from Navier-Stokes. Those are unique and stable among inviscid limits from Navier-Stokes. The proof is based on the three main methodologies: the modified front tracking algorithm; the a-contraction with shifts; the method of compensated compactness. In the first lecture, I will introduce the method of a-contraction with shifts. In the second lecture, I will present the proof for the well-posedness of Cauchy problem.

일시: 2024년 2월 19일 (월) 16:00~17:00

장소: 자연과학대학 213호

연사: 이현주 연구원 (연세대학교의료원 영상의학과)

주제: Decision-making approach to enhance the performance of neural networks for medical image classification

초록:  In this study, we propose a simple but effective method to improve the performance of transfer learning-based networks for medical imaging on given data. The proposed method has two steps. The Original network is used for initial decision-making, and some of its decisions are passed to a second decision-making stage involving a new network called the True network. A True network has the same architecture as the Original network but is trained only on a subset of the original data that meets certain conditions. It is then used to validate the classification results of the Original network, and misclassified images are identified and reclassified. The effectiveness of this approach was evaluated on thyroid nodule ultrasound images using the ResNet101 architecture as the representative network. The proposed method enhanced performance in all five metrics, including accuracy, sensitivity, specificity, f1-score, and AUC compared to using only the Original network or True network. Additional experiments showed improved performance of several other CNNs for the medical images as well. Our proposed method improves the performance of transfer learning-based networks in medical imaging without additional data, even when processing limited amounts of data. The approach proposed can potentially increase the accuracy of medical image classification, making it a promising technology for medical image analysis.

일시: 2024년 2월 19일 (월) 16:00~17:00

장소: 자연과학대학 213호

연사: 김수현 박사님 (부산대학교 수학과)

주제: Gauss–Legendre polynomials for parametric curve shape adjustment

초록: Controlling the deformation of polynomial curves has significant importance in various domains, including computer graphics and computer-aided design (CAM/CAD/CAGD). One widely-used expression form of a poly-nomial curve is the B´ezier curve, which is constructed by barycentric combination of the Bernstein polynomial basis with the given control points. In this presentation, we will introduce the Gauss–Legendre polynomial, originally derived from the process of developing as the rectifying control polygon of Pythagorean hodograph(PH) curves. Gauss-Legendre polynomials have the same length as the arc length of the corresponding PH curve. If this concept is extended to a polynomial curve, the shape of the curve can be easily predicted and adjusted using the proposed control polygon, even for high-degree polynomial curves. Similar to how a B´ezier curve relies on Bernstein polynomials for its expression, a PH curve can be expressed through a linear combi-nation of Gauss–Legendre control points, with coefficients determined by Gauss–Legendre polynomials. These polynomials can be constructed from Lagrange interpolators defined over the roots of Legendre polynomials. This research is joint work with Hwan Pyo Moon and Song-Hwa Kwon.

일시:  2023년 09월 04일 (월) 17:00~18:00

장소:  경북대학교 자연과학대학 319호

연사:  Masahiro Suzuki 교수님(Nagoya Institute of Technology)

주제: Solitary waves of the Vlasov-Poisson system

초록:  We consider the Vlasov-Poisson system describing a two-species plasma with spatial dimension 1 and the velocity variable in $\mathbb{R}^n$. We find the necessary and sufficient conditions for the existence of solitary waves of the system. To this end, we need to investigate the distribution of ions trapped by the electrostatic potential. Furthermore, we classify completely in all possible cases whether or not the solitary wave is unique, when we exclude the variant caused by translation. There are both cases that it is unique and nonunique.