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Hyung Ju Hwang

Publications and source records attributed to Hyung Ju Hwang.

At least 37 records · Page 2Linked to original sources
arXiv2021

Lagrangian dual framework for conservative neural network solutions of kinetic equations

In this paper, we propose a novel conservative formulation for solving kinetic equations via neural networks. More precisely, we formulate the learning problem as a constrained optimization problem with constraints that represent the physical conservation laws. The constraints are relaxed toward the residual loss function by the Lagrangian duality. By imposing physical conservation properties of the solution as constraints of the learning problem, we demonstrate far more accurate approximations of the solutions in terms of errors and the conservation laws, for the kinetic Fokker-Planck equation and the homogeneous Boltzmann equation.

math.NA↗
arXiv2021

Option Compatible Reward Inverse Reinforcement Learning

Reinforcement learning in complex environments is a challenging problem. In particular, the success of reinforcement learning algorithms depends on a well-designed reward function. Inverse reinforcement learning (IRL) solves the problem of recovering reward functions from expert demonstrations. In this paper, we solve a hierarchical inverse reinforcement learning problem within the options framework, which allows us to utilize intrinsic motivation of the expert demonstrations. A gradient method for parametrized options is used to deduce a defining equation for the Q-feature space, which leads to a reward feature space. Using a second-order optimality condition for option parameters, an optimal reward function is selected. Experimental results in both discrete and continuous domains confirm that our recovered rewards provide a solution to the IRL problem using temporal abstraction, which in turn are effective in accelerating transfer learning tasks. We also show that our method is robust to noises contained in expert demonstrations.

cs.LG↗
arXiv2020

$L^2$ decay for the linearized Landau equation with the specular boundary condition

In this paper, we develop an alternative approach to establish the $L^2$ decay estimate for the linearized Landau equation in a bounded domain with specular boundary condition. The proof is based on the methodology of proof by contradiction motivated by [Guo, Comm. Pure Appl. Math., 55(9):1104-1135, 2002] and [Guo, Arch. Ration. Mech. Anal., 197(3):713-809, 2010].

math.AP↗
arXiv2020

The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach

The model reduction of a mesoscopic kinetic dynamics to a macroscopic continuum dynamics has been one of the fundamental questions in mathematical physics since Hilbert's time. In this paper, we consider a diagram of the diffusion limit from the Vlasov-Poisson-Fokker-Planck (VPFP) system on a bounded interval with the specular reflection boundary condition to the Poisson-Nernst-Planck (PNP) system with the no-flux boundary condition. We provide a Deep Learning algorithm to simulate the VPFP system and the PNP system by computing the time-asymptotic behaviors of the solution and the physical quantities. We analyze the convergence of the neural network solution of the VPFP system to that of the PNP system via the Asymptotic-Preserving (AP) scheme. Also, we provide several theoretical evidence that the Deep Neural Network (DNN) solutions to the VPFP and the PNP systems converge to the a priori classical solutions of each system if the total loss function vanishes.

math.NA↗
arXiv2020

Posting Bot Detection on Blockchain-based Social Media Platform using Machine Learning Techniques

Steemit is a blockchain-based social media platform, where authors can get author rewards in the form of cryptocurrencies called STEEM and SBD (Steem Blockchain Dollars) if their posts are upvoted. Interestingly, curators (or voters) can also get rewards by voting others' posts, which is called a curation reward. A reward is proportional to a curator's STEEM stakes. Throughout this process, Steemit hopes "good" content will be automatically discovered by users in a decentralized way, which is known as the Proof-of-Brain (PoB). However, there are many bot accounts programmed to post automatically and get rewards, which discourages real human users from creating good content. We call this type of bot a posting bot. While there are many papers that studied bots on traditional centralized social media platforms such as Facebook and Twitter, we are the first to study posting bots on a blockchain-based social media platform. Compared with the bot detection on the usual social media platforms, the features we created have an advantage that posting bots can be detected without limiting the number or length of posts. We can extract the features of posts by clustering distances between blog data or replies. These features are obtained from the Minimum Average Cluster from Clustering Distance between Frequent words and Articles (MAC-CDFA), which is not used in any of the previous social media research. Based on the enriched features, we enhanced the quality of classification tasks. Comparing the F1-scores, the features we created outperformed the features used for bot detection on Facebook and Twitter.

cs.SI↗
arXiv2019

Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach

The issue of the relaxation to equilibrium has been at the core of the kinetic theory of rarefied gas dynamics. In the paper, we introduce the Deep Neural Network (DNN) approximated solutions to the kinetic Fokker-Planck equation in a bounded interval and study the large-time asymptotic behavior of the solutions and other physically relevant macroscopic quantities. We impose the varied types of boundary conditions including the inflow-type and the reflection-type boundaries as well as the varied diffusion and friction coefficients and study the boundary effects on the asymptotic behaviors. These include the predictions on the large-time behaviors of the pointwise values of the particle distribution and the macroscopic physical quantities including the total kinetic energy, the entropy, and the free energy. We also provide the theoretical supports for the pointwise convergence of the neural network solutions to the \textit{a priori} analytic solutions. We use the library \textit{PyTorch}, the activation function \textit{tanh} between layers, and the \textit{Adam} optimizer for the Deep Learning algorithm.

math.NA↗
arXiv2019

Deep Neural Network Approach to Forward-Inverse Problems

In this paper, we construct approximated solutions of Differential Equations (DEs) using the Deep Neural Network (DNN). Furthermore, we present an architecture that includes the process of finding model parameters through experimental data, the inverse problem. That is, we provide a unified framework of DNN architecture that approximates an analytic solution and its model parameters simultaneously. The architecture consists of a feed forward DNN with non-linear activation functions depending on DEs, automatic differentiation, reduction of order, and gradient based optimization method. We also prove theoretically that the proposed DNN solution converges to an analytic solution in a suitable function space for fundamental DEs. Finally, we perform numerical experiments to validate the robustness of our simplistic DNN architecture for 1D transport equation, 2D heat equation, 2D wave equation, and the Lotka-Volterra system.

math.NA↗
arXiv2019

The Landau equation with the specular reflection boundary condition

The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight of this work also comes from the low-regularity assumptions made for the initial distribution. This work generalizes the recent global stability result for the Landau equation in a periodic box (KGH-2016). Our methods consist of the generalization of the wellposedness theory for the Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (Polidoro-Ragusa-1998) to further control the velocity derivatives, which ensures the uniqueness. Our methods provide a new understanding of the grazing collisions in the Landau theory for an initial-boundary value problem.

math.AP↗
arXiv2018

Local Stability and Performance of Simple Gradient Penalty mu-Wasserstein GAN

Wasserstein GAN(WGAN) is a model that minimizes the Wasserstein distance between a data distribution and sample distribution. Recent studies have proposed stabilizing the training process for the WGAN and implementing the Lipschitz constraint. In this study, we prove the local stability of optimizing the simple gradient penalty $μ$-WGAN(SGP $μ$-WGAN) under suitable assumptions regarding the equilibrium and penalty measure $μ$. The measure valued differentiation concept is employed to deal with the derivative of the penalty terms, which is helpful for handling abstract singular measures with lower dimensional support. Based on this analysis, we claim that penalizing the data manifold or sample manifold is the key to regularizing the original WGAN with a gradient penalty. Experimental results obtained with unintuitive penalty measures that satisfy our assumptions are also provided to support our theoretical results.

cs.LG↗
arXiv2018

Nonuniqueness for the kinetic Fokker-Planck equation with inelastic boundary conditions

We describe the structure of solutions of the kinetic Fokker-Planck equations in domains with boundaries near the singular set in one-space dimension. We study in particular the behaviour of the solutions of this equation for inelastic boundary conditions which are characterized by means of a coefficient $r$ describing the amount of energy lost in the collisions of the particles with the boundaries of the domain. A peculiar feature of this problem is the onset of a critical exponent rc which follows from the analysis of McKean (cf. [26]) of the properties of the stochastic process associated to the Fokker-Planck equation under consideration. In this paper, we prove rigorously that the solutions of the considered problem are nonunique if $r < r_c$ and unique if $r_{c}<r\leq 1$. In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the Fokker-Planck equation. In the proof of the results of this paper we use several asymptotic formulas and computations in the companion paper [16].

math.AP↗
arXiv2018

On the structure of the singular set for the kinetic Fokker-Planck equations in domains with boundaries

In this paper we compute asymptotics of solutions of the kinetic Fokker-Planck equation with inelastic boundary conditions which indicate that the solutions are nonunique if $r < r_c$. The nonuniqueness is due to the fact that different solutions can interact in a different manner with a Dirac mass which appears at the singular point $(x,v)=(0,0)$. In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the kinetic Fokker-Planck equation. The asymptotics obtained in this paper will be used in a companion paper [34] to prove rigorously nonuniqueness of solutions for the kinetic Fokker-Planck equation with inelastic boundary conditions.

math.AP↗
arXiv2017

The effect of time-varying flow-shear on the nonlinear stability of the boundary of magnetized toroidal plasmas

We propose a phenomenological yet very general model in a form of generalized complex Ginzburg-Landau equation to understand the dynamics of the quasi-periodic fluid instabilities (called edge-localized modes) in the boundary of toroidal magnetized high-temperature plasmas. The model reproduces key dynamical features of the boundary instabilities observed in the high-confinement state plasmas on the KSTAR tokamak, including quasi-steady states characterized by field-aligned filamentary eigenmodes, transitions between different eigenmodes, and rapid transition to non-modal filamentary structure prior to the relaxation. It is found that the inclusion of time-varying perpendicular sheared flow is crucial for reproducing the observed dynamical features.

physics.plasm-ph↗
arXiv2017

Mathematical analysis of long-time behavior of magnetized fluid instabilities with shear flow

We study a complex Ginzburg-Landau (GL) type model related to fluid instabilities in the boundary of magnetized toroidal plasmas (called edge-localized modes) with a prescribed shear flow on the Neumann boundary condition. We obtain the following universal results for the model in a one-dimensional interval. First, if the shear is weak, there is a unique linearly stable steady-state perturbed from the nonzero constant steady-state corresponding to the zero shear case. Second, if the shear is strong, there is no plausible steady-state except the trivial zero solution in the interval. With the help of these results and the existence of global attractors, we can dramatically reduce the number of cases for the long-time behavior of a solution in the model.

physics.plasm-ph↗
arXiv2016

A $L^{2}$ to $L^{\infty}$ approach for the Landau Equation

Consider the Landau equation with Coulomb potential in a periodic box. We develop a new $L^{2}\rightarrow L^{\infty }$ framework to construct global unique solutions near Maxwellian with small $L^{\infty }\ $norm. The first step is to establish global $L^{2}$ estimates with strong velocity weight and time decay, under the assumption of $L^{\infty }$ bound, which is further controlled by such $L^{2}$ estimates via De Giorgi's method \cite{golse2016harnack} and \cite{mouhot2015holder}. The second step is to employ estimates in $S_{p}$ spaces to control velocity derivatives to ensure uniqueness, which is based on Holder estimates via De Giorgi's method \cite{golse2016harnack}, \cite{golse2015holder}, and \cite{mouhot2015holder}.

math.AP↗
arXiv2014

A stable self-similar singularity of evaporating drops: ellipsoidal collapse to a point

We study the problem of evaporating drops contracting to a point. Going back to Maxwell and Langmuir, the existence of a spherical solution for which evaporating drops collapse to a point in a self-similar manner is well established in the physical literature. The diameter of the drop follows the so-called $D^{2}$ law: the second power of the drop-diameter decays linearly in time. In this study we provide a complete mathematical proof of this classical law. We prove that evaporating drops which are initially small perturbations of a sphere collapse to a point and the shape of the drop converges to a self-similar ellipsoid whose center, orientation, and semi-axes are determined by the initial shape.

math.AP↗
arXiv2013

THE Fokker-Planck Equation With Absorbing Boundary Conditions

We study the initial-boundary value problem for the Fokker-Planck equation in an interval with absorbing boundary conditions. We develop a theory of well-posedness of classical solutions for the problem. We also prove that the resulting solutions decay exponentially for long times. To prove these results we obtain several crucial estimates, which include hypoellipticity away from the singular set for the Fokker-Planck equation with absorbing boundary conditions, as well as the Holder continuity of the solutions up to the singular set.

math.AP↗
arXiv2011

On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian

We establish the exponential time decay rate of smooth solutions of small amplitude to the Vlasov-Poisson-Fokker-Planck equations to the Maxwellian both in the whole space and in the periodic box via the uniform-in-time energy estimates and also the macroscopic equations.

math.AP↗