1. Time: December 22nd (Monday) 10:00
Place: Natural Science Building #744
Speaker: Dr. Jaeyoung Yoon (Technical University of Munich)

Title: Stochastic Modified Equations for SGD in Infinite-Dimensional Hilbert Spaces

Abstract:
Stochastic Gradient Descent (SGD) is a simple and computationally efficient algorithm, but its discrete and non-Gaussian stochastic nature makes a theoretical analysis of its dynamics challenging. To address this difficulty, we formulate a stochastic modified equation (SME) in an infinite-dimensional Hilbert space that approximates SGD in continuous time. We show that the SME accurately captures the expected behavior of SGD for a wide class of functionals, and that non-Gaussian mini-batch noise can be effectively approximated by Gaussian noise in this continuous-time model. Through numerical experiments, we illustrate how the difference between SGD and its continuous-time SME approximation depends on the step size, of which results correspond to the weak convergence analysis. Our work extends Weinan E’s stochastic modified equation theory from finite-dimensional setting ([1]) to infinite-dimensional Hilbert spaces.

[1] Li, Qianxiao, Cheng Tai, and Weinan E. “Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations.” Journal of Machine Learning Research 20 (40): 1–47, 2019.

2. Time: December 22nd (Monday) 11:00
Place: Natural Science Building #744
Speaker: Prof. Junhyeok Byeon (Dalian University of Technology)

Title: A stochastic consensus model for global optimization

Abstract:
We propose a first-order, time-discrete stochastic consensus model for global optimization. The model draws on interaction-based mechanisms to incorporate objective-function information and handles non-convex, non-differentiable, and even discontinuous functions. It is motivated by the Consensus-Based Optimization (CBO) paradigm, which promotes consensus among agents toward a global optimum through simple stochastic dynamics amenable to rigorous mathematical analysis. Despite these promises, the actual behavior of agents in its time-discrete implementation remains largely unknown. We address this issue by the novel observation that the consensus point governs the entire ensemble. We further demonstrate competitive performance across various problems.

[Link to Young PDE 5 workshop homepage]

– Date: January 14-16, 2026

– Place: Hanyang University, Seoul

– Participation/Attendence Form
[Link to Google Form]

– Speakers
Gayoung An (Yonsei University)
Jae-Hwan Choi (KIAS)
Jin Woo Jang (POSTECH)
Seongmin Jeon (Hanyang University)
Yong-Gwan Ji (KIAS)
Sangdon Jin (Chungbuk National University)
Dong-ha Kim (Ajou University)
DongKwang Kim (UNIST)
Jinyeop Lee (University of Basel)
Se-Chan Lee (KIAS)
Woojae Lee (Yonsei University)
Deokwoo Lim (Seoul National University)
Bora Moon (Yonsei University)
Jehan Oh (Kyungpook National University)
Jinsol Seo (KIAS)
Hansol Park (National Tsing Hua University)
Kihoon Seong (MSRI/SLMath)
Jaeyong Shin (Yonsei University)
Hyungsung Yun (Changwon National University)

– Organizers
Jinwook Jung (Hanyang University)
Junha Kim (Ajou University)
Minhyun Kim (Hanyang University)
Seunghyeok Kim (Hanyang University)
Jinyeong Park (Hanyang University)

– Support
Department of Mathematics, Hanyang University
BK21 AJOU Mathematical Sciences Team for Future Leaders, Ajou University

[Young PDEs main page]

1. Time: October 21st (Tuesday) 10:00
Place: Natural Science Building #744
Speaker: Dr. Gyuyoung Hwang (Institute for Basic Science)

Title: Forward and inverse problems in coupled oscillator systems

Abstract:
When oscillators interact each other, they exhibit variety of phenomenon such as synchronization, chaotic motion, and kimera states. Such a system, called coupled oscillator system, is prevalent from neuronal oscillators and circadian rhythms to quantum optics and engineering. In this talk, we study the forward and inverse problem of the coupled oscillator systems. For the forward problem, we investigate the existence of periodic measure-valued solutions for a class of nonlocal continuity equations, which include the mean-field equations arising from the coupled systems of oscillators. We use a fixed point theorem for geodesically convex spaces constructed by optimal mass transport. On the other hand, in many real-life situations, we often lack the information of the exact model, in particular the interaction force between oscillators. Thus, in the next section, we study the inverse problem to search for the coupling function among oscillators using physics-informed neural networks.

2. Time: October 21st (Tuesday) 11:00
Place: Natural Science Building #744
Speaker: Prof. Jeongho Kim (Kyung Hee University)

Title: Asymptotic behavior of the Navier-Stokes-Korteweg system

Abstract:
Fluids with capillary effects appear in a wide range of contexts, including multiphase flow and quantum hydrodynamics. In this talk, I will introduce the compressible Navier–Stokes–Korteweg equations, which are extensions of the standard compressible fluid models that incorporate capillarity. Then, I will discuss the stability of viscous–dispersive shock profiles, and its composition with the rarefaction wave for the compressible Navier–Stokes–Korteweg system. These results are based on joint work with Prof. Moon-Jin Kang and Dr. Sungho Han (KAIST).

Time: June 18th (Wednesday) 11:00
Place: Natural Science Building #744
Speaker: Dr. Junhwa Jung (Brown University)

Title: Diffusive limit of the Boltzmann equation with boundary conditions

Abstract:
The derivation of fluid equations from the Boltzmann equation is one of the most important problems in kinetic theory. In this talk, we investigate several results concerning the diffusive limit of the Boltzmann equation within the \(L^2 – L^\infty\) framework. Based on this framework, we present two results: (1) a global diffusive expansion in an exterior domain, and (2) the global hydrodynamic limit with Maxwell boundary conditions in a bounded domain.

1. Time: April 14th (Monday) 10:00
Place: Natural Science Building #744
Speaker: Dr. Jaeyoung Yoon (Technical University of Munich)

Title: Adaptive Cucker-Smale Model and its Asymptotic Behavior in the Singular Limit

Abstract:
In this talk, we introduce an adaptive network within Cucker-Smale (CS) dynamics. The properties of adaptive networks allow particles in the CS regime to form and break up groups of neighbors, resulting in the emergence of diverse patterns. We investigate the singular limit of the adaptive CS model to better understand the role of the adaptive rule, which transforms the system into Laplacian dynamics on a temporal graph. Through the analysis of Laplacian dynamics on various types of temporal graphs, we demonstrate the asymptotic behavior of the adaptive CS model in this singular limit.

2. Time: April 14th (Monday) 11:00
Place: Natural Science Building #744
Speaker: Youngseok Yoo (Yonsei University)

Title: Existence and Large-time behavior of the solutions for the nonlinear Vlasov-Poisson-Fokker-Planck Equation

Abstract:
The Vlasov equation models collisionless systems with long-range interactions, such as plasmas or galaxies, while the Fokker-Planck equation accounts for random forces including thermal fluctuations, collisions, and noise. The Vlasov-Fokker-Planck equation describes the evolution of a particle distribution function in phase space under the influence of self-consistent forces, external potentials, and diffusive effects. This framework is fundamental in fields such as plasma physics, kinetic theory, and statistical mechanics. In this talk, we focus on the nonlinear Vlasov-Poisson-Fokker-Planck equation, where the force arises from self-consistent interactions, meaning that the particles generate a field that, in turn, acts back on the system. We study the existence and large-time behavior of solutions to this equation in the whole space. We first establish the local existence and uniqueness of solutions for suitably regular initial data. These results are then extended to global existence through the use of energy estimates and macro-micro decomposition techniques. Finally, we analyze the long-time behavior of solutions, demonstrating algebraic decay toward equilibrium under appropriate conditions.

Date: 22nd August (Thursday), 2024
Place: #702, Natural Science Building, Hanyang University

HYKE2024 program(pdf)

Time: July 25nd (Thursday) 10:30
Place: Natural Science Building #744
Speaker: Prof. Woojoo Shim (Kyungpook National University)

Title: Synchronization estimate of the discrete Kuramoto model with multiplicative random noise

Abstract:
In this talk, we study the emergence of synchronization of time-discretization of the Kuramoto model with multiplicative noise. First, we will briefly describe several methods for approximating the solution of stochastic differential equations, including the method we used in the paper. Then, we introduce the existing results of the continuous-time stochastic Kuramoto model and the discrete-time deterministic Kuramoto model. Finally, we will discuss a sufficient condition that guarantees the identical oscillators in the discrete-time Kuramoto model with multiplicative noise are asymptotically synchronized almost surely.