arXiv subjects

Jae-Hun Jung

Publications and source records attributed to Jae-Hun Jung.

At least 37 records · Page 2Linked to original sources

arXiv2024

Neural network Approximations for Reaction-Diffusion Equations -- Homogeneous Neumann Boundary Conditions and Long-time Integrations

Reaction-Diffusion systems arise in diverse areas of science and engineering. Due to the peculiar characteristics of such equations, analytic solutions are usually not available and numerical methods are the main tools for approximating the solutions. In the last decade, artificial neural networks have become an active area of development for solving partial differential equations. However, several challenges remain unresolved with these methods when applied to reaction-diffusion equations. In this work, we focus on two main problems. The implementation of homogeneous Neumann boundary conditions and long-time integrations. For the homogeneous Neumann boundary conditions, we explore four different neural network methods based on the PINN approach. For the long time integration in Reaction-Diffusion systems, we propose a domain splitting method in time and provide detailed comparisons between different implementations of no-flux boundary conditions. We show that the domain splitting method is crucial in the neural network approach, for long time integration in Reaction-Diffusion systems. We demonstrate numerically that domain splitting is essential for avoiding local minima, and the use of different boundary conditions further enhances the splitting technique by improving numerical approximations. To validate the proposed methods, we provide numerical examples for the Diffusion, the Bistable and the Barkley equations and provide a detailed discussion and comparisons of the proposed methods.

math.NA↗

arXiv2024

A Numerical Study of WENO Approximations to Sharp Propagating Fronts for Reaction-Diffusion Systems

Many reaction-diffusion systems in various applications exhibit traveling wave solutions that evolve on multiple spatio-temporal scales. These traveling wave solutions are crucial for understanding the underlying dynamics of the system. In this work, we present sixth-order weighted essentially non-oscillatory (WENO) methods within the finite difference framework to solve reaction-diffusion systems. The WENO method allows us to use fewer grid points and larger time steps compared to classical finite difference methods. Our focus is on solving the reaction-diffusion system for the traveling wave solution with the sharp front. Although the WENO method is popular for hyperbolic conservation laws, especially for problems with discontinuity, it can be adapted for the equations of parabolic type, such as reaction-diffusion systems, to effectively handle sharp wave fronts. Thus, we employed the WENO methods specifically developed for equations of parabolic type. We considered various reaction-diffusion equations, including Fisher's, Zeldovich, Newell-Whitehead-Segel, bistable equations, and the Lotka-Volterra competition-diffusion system, all of which yield traveling wave solutions with sharp wave fronts. Numerical examples in this work demonstrate that the central WENO method is highly more accurate and efficient than the commonly used finite difference method. We also provide an analysis related to the numerical speed of the sharp propagating front in the Newell-Whitehead-Segel equation. The overall results confirm that the central WENO method is highly efficient and is recommended for solving reaction-diffusion equations with sharp wave fronts.

math.NA↗

arXiv2024

A third-order finite difference weighted essentially non-oscillatory scheme with shallow neural network

In this paper, we introduce the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. We employ the supervised learning and design two loss functions, one with the mean squared error and the other with the mean squared logarithmic error, where the WENO3-JS weights are computed as the labels. Each loss function consists of two components where the first component compares the difference between the weights from the neural network and WENO3-JS weights, while the second component matches the output weights of the neural network and the linear weights. The former of the loss function enforces the neural network to follow the WENO properties, implying that there is no need for the post-processing layer. Additionally the latter leads to better performance around discontinuities. As a neural network structure, we choose the shallow neural network (SNN) for computational efficiency with the Delta layer consisting of the normalized undivided differences. These constructed WENO3-SNN schemes show the outperformed results in one-dimensional examples and improved behavior in two-dimensional examples, compared with the simulations from WENO3-JS and WENO3-Z.

cs.LG↗

arXiv2024

Supervised low-rank semi-nonnegative matrix factorization with frequency regularization for forecasting spatio-temporal data

We propose a novel methodology for forecasting spatio-temporal data using supervised semi-nonnegative matrix factorization (SSNMF) with frequency regularization. Matrix factorization is employed to decompose spatio-temporal data into spatial and temporal components. To improve clarity in the temporal patterns, we introduce a nonnegativity constraint on the time domain along with regularization in the frequency domain. Specifically, regularization in the frequency domain involves selecting features in the frequency space, making an interpretation in the frequency domain more convenient. We propose two methods in the frequency domain: soft and hard regularizations, and provide convergence guarantees to first-order stationary points of the corresponding constrained optimization problem. While our primary motivation stems from geophysical data analysis based on GRACE (Gravity Recovery and Climate Experiment) data, our methodology has the potential for wider application. Consequently, when applying our methodology to GRACE data, we find that the results with the proposed methodology are comparable to previous research in the field of geophysical sciences but offer clearer interpretability.

stat.ML↗

arXiv2024

Stratified Sampling Algorithms for Machine Learning Methods in Solving Two-scale Partial Differential Equations

Partial differential equations (PDEs) with multiple scales or those defined over sufficiently large domains arise in various areas of science and engineering and often present problems when approximating the solutions numerically. Machine learning techniques are a relatively recent method for solving PDEs. Despite the increasing number of machine learning strategies developed to approximate PDEs, many remain focused on relatively small domains. When scaling the equations, a large domain is naturally obtained, especially when the solution exhibits multiscale characteristics. This study examines two-scale equations whose solution structures exhibit distinct characteristics: highly localized in some regions and significantly flat in others. These two regions must be adequately addressed over a large domain to approximate the solution more accurately. We focus on the vanishing gradient problem given by the diminishing gradient zone of the activation function over large domains and propose a stratified sampling algorithm to address this problem. We compare the uniform random classical sampling method over the entire domain and the proposed stratified sampling method. The numerical results confirm that the proposed method yields more accurate and consistent solutions than classical methods.

math.NA↗

arXiv2024

Persistent homology of featured time series data and its applications

Recent studies have actively employed persistent homology (PH), a topological data analysis technique, to analyze the topological information in time series data. Many successful studies have utilized graph representations of time series data for PH calculation. Given the diverse nature of time series data, it is crucial to have mechanisms that can adjust the PH calculations by incorporating domain-specific knowledge. In this context, we introduce a methodology that allows the adjustment of PH calculations by reflecting relevant domain knowledge in specific fields. We introduce the concept of featured time series, which is the pair of a time series augmented with specific features such as domain knowledge, and an influence vector that assigns a value to each feature to fine-tune the results of the PH. We then prove the stability theorem of the proposed method, which states that adjusting the influence vectors grants stability to the PH calculations. The proposed approach enables the tailored analysis of a time series based on the graph representation methodology, which makes it applicable to real-world domains. We consider two examples to verify the proposed method's advantages: anomaly detection of stock data and topological analysis of music data.

math.AT↗

arXiv2024

A spatial-temporal weight analysis and novel nonlinear weights of weighted essentially non-oscillatory schemes for hyperbolic conservation laws

In this paper we analyze the weighted essentially non-oscillatory (WENO) schemes in the finite volume framework by examining the first step of the explicit third-order total variation diminishing Runge-Kutta method. The rationale for the improved performance of the finite volume WENO-M, WENO-Z and WENO-ZR schemes over WENO-JS in the first time step is that the nonlinear weights corresponding to large errors are adjusted to increase the accuracy of numerical solutions. Based on this analysis, we propose novel Z-type nonlinear weights of the finite volume WENO scheme for hyperbolic conservation laws. Instead of taking the difference of the smoothness indicators for the global smoothness indicator, we employ the logarithmic function with tuners to ensure that the numerical dissipation is reduced around discontinuities while the essentially non-oscillatory property is preserved. The proposed scheme does not necessitate substantial extra computational expenses. Numerical examples are presented to demonstrate the capability of the proposed WENO scheme in shock capturing.

math.NA↗

arXiv2024

Explicit radial basis function Runge-Kutta methods

The aim of this paper is to design the explicit radial basis function (RBF) Runge-Kutta methods for the initial value problem. We construct the two-, three- and four-stage RBF Runge-Kutta methods based on the Gaussian RBF Euler method with the shape parameter, where the analysis of the local truncation error shows that the s-stage RBF Runge-Kutta method could formally achieve order s+1. The proof for the convergence of those RBF Runge-Kutta methods follows. We then plot the stability region of each RBF Runge-Kutta method proposed and compare with the one of the correspondent Runge-Kutta method. Numerical experiments are provided to exhibit the improved behavior of the RBF Runge-Kutta methods over the standard ones.

math.NA↗

arXiv2024

Weighted Isolation and Random Cut Forest Algorithms for Anomaly Detection

Random cut forest (RCF) algorithms have been developed for anomaly detection, particularly in time series data. The RCF algorithm is an improved version of the isolation forest (IF) algorithm. Unlike the IF algorithm, the RCF algorithm can determine whether real-time input contains an anomaly by inserting the input into the constructed tree network. Various RCF algorithms, including Robust RCF (RRCF), have been developed, where the cutting procedure is adaptively chosen probabilistically. The RRCF algorithm demonstrates better performance than the IF algorithm, as dimension cuts are decided based on the geometric range of the data, whereas the IF algorithm randomly chooses dimension cuts. However, the overall data structure is not considered in both IF and RRCF, given that split values are chosen randomly. In this paper, we propose new IF and RCF algorithms, referred to as the weighted IF (WIF) and weighted RCF (WRCF) algorithms, respectively. Their split values are determined by considering the density of the given data. To introduce the WIF and WRCF, we first present a new geometric measure, a density measure, which is crucial for constructing the WIF and WRCF. We provide various mathematical properties of the density measure, accompanied by theorems that support and validate our claims through numerical examples.

cs.LG↗

arXiv2022

Fifth-order weighted essentially non-oscillatory schemes with new Z-type nonlinear weights for hyperbolic conservation laws

In this paper we propose new Z-type nonlinear weights of the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme for hyperbolic conservation laws. Instead of employing the classical smoothness indicators for the nonlinear weights, we take the $p$th root of the smoothness indicators and follow the form of Z-type nonlinear weights, leading to fifth order accuracy in smooth regions, even at the critical points, and sharper approximations around the discontinuities. We also prove that the proposed nonlinear weights converge to the linear weights as $p \to \infty$, implying the convergence of the resulting WENO numerical flux to the finite difference numerical flux. Finally, numerical examples are presented by comparing with other WENO schemes, such as WENO-JS, WENO-M and WENO-Z, to demonstrate that the proposed WENO scheme performs better in shock capturing.

math.NA↗

arXiv2022

Machine Composition of Korean Music via Topological Data Analysis and Artificial Neural Network

Common AI music composition algorithms based on artificial neural networks are to train a machine by feeding a large number of music pieces and create artificial neural networks that can produce music similar to the input music data. This approach is a blackbox optimization, that is, the underlying composition algorithm is, in general, not known to users. In this paper, we present a way of machine composition that trains a machine the composition principle embedded in the given music data instead of directly feeding music pieces. We propose this approach by using the concept of {\color{black}{Overlap}} matrix proposed in \cite{TPJ}. In \cite{TPJ}, a type of Korean music, so-called the {\it Dodeuri} music such as Suyeonjangjigok has been analyzed using topological data analysis (TDA), particularly using persistent homology. As the raw music data is not suitable for TDA analysis, the music data is first reconstructed as a graph. The node of the graph is defined as a two-dimensional vector composed of the pitch and duration of each music note. The edge between two nodes is created when those nodes appear consecutively in the music flow. Distance is defined based on the frequency of such appearances. Through TDA on the constructed graph, a unique set of cycles is found for the given music. In \cite{TPJ}, the new concept of the {\it {\color{black}{Overlap}} matrix} has been proposed, which visualizes how those cycles are interconnected over the music flow, in a matrix form. In this paper, we explain how we use the {\color{black}{Overlap}} matrix for machine composition. The {\color{black}{Overlap}} matrix makes it possible to compose a new music piece algorithmically and also provide a seed music towards the desired artificial neural network. In this paper, we use the {\it Dodeuri} music and explain detailed steps.

cs.SD↗

arXiv2021

Instability of the Betti Sequence for Persistent Homology and a Stabilized Version of the Betti Sequence

Topological Data Analysis (TDA), a relatively new field of data analysis, has proved very useful in a variety of applications. The main persistence tool from TDA is persistent homology in which data structure is examined at many scales. Representations of persistent homology include persistence barcodes and persistence diagrams, both of which are not straightforward to reconcile with traditional machine learning algorithms as they are sets of intervals or multisets. The problem of faithfully representing barcodes and persistent diagrams has been pursued along two main avenues: kernel methods and vectorizations. One vectorization is the Betti sequence, or Betti curve, derived from the persistence barcode. While the Betti sequence has been used in classification problems in various applications, to our knowledge, the stability of the sequence has never before been discussed. In this paper we show that the Betti sequence is unstable under the 1-Wasserstein metric with regards to small perturbations in the barcode from which it is calculated. In addition, we propose a novel stabilized version of the Betti sequence based on the Gaussian smoothing seen in the Stable Persistence Bag of Words for persistent homology. We then introduce the normalized cumulative Betti sequence and provide numerical examples that support the main statement of the paper.

math.AT↗

arXiv2021

Topological Data Analysis of Korean Music in Jeongganbo: A Cycle Structure

Jeongganbo is a unique music representation invented by Sejong the Great. Contrary to the western music notation, the pitch of each note is encrypted and the length is visualized directly in a matrix form in Jeongganbo. We use topological data analysis (TDA) to analyze the Korean music written in Jeongganbo for Suyeonjang, Songuyeo, and Taryong, those well-known pieces played at the palace and among noble community. We are particularly interested in the cycle structure. We first define and determine the node elements of each music, characterized uniquely with its pitch and length. Then we transform the music into a graph and define the distance between the nodes as their adjacent occurrence rate. The graph is used as a point cloud whose homological structure is investigated by measuring the hole structure in each dimension. We identify cycles of each music, match those in Jeongganbo, and show how those cycles are interconnected. The main discovery of this work is that the cycles of Suyeonjang and Songuyeo, categorized as a special type of cyclic music known as Dodeuri, frequently overlap each other when appearing in the music while the cycles found in Taryong, which does not belong to Dodeuri class, appear individually.

cs.SD↗

arXiv2021

A Chebyshev multidomain adaptive mesh method for Reaction-Diffusion equations

Reaction-Diffusion equations can present solutions in the form of traveling waves. Such solutions evolve in different spatial and temporal scales and it is desired to construct numerical methods that can adopt a spatial refinement at locations with large gradient solutions. In this work we develop a high order adaptive mesh method based on Chebyshev polynomials with a multidomain approach for the traveling wave solutions of reaction-diffusion systems, where the proposed method uses the non-conforming and non-overlapping spectral multidomain method with the temporal adaptation of the computational mesh. Contrary to the existing multidomain spectral methods for reaction-diffusion equations, the proposed multidomain spectral method solves the given PDEs in each subdomain locally first and the boundary and interface conditions are solved in a global manner. In this way, the method can be parallelizable and is efficient for the large reaction-diffusion system. We show that the proposed method is stable and provide both the one- and two-dimensional numerical results that show the efficacy of the proposed method.

math.NA↗

arXiv2020

The Interconnectivity Vector: A Finite-Dimensional Vector Representation of Persistent Homology

Persistent Homology (PH) is a useful tool to study the underlying structure of a data set. Persistence Diagrams (PDs), which are 2D multisets of points, are a concise summary of the information found by studying the PH of a data set. However, PDs are difficult to incorporate into a typical machine learning workflow. To that end, two main methods for representing PDs have been developed: kernel methods and vectorization methods. In this paper we propose a new finite-dimensional vector, called the interconnectivity vector, representation of a PD adapted from Bag-of-Words (BoW). This new representation is constructed to demonstrate the connections between the homological features of a data set. This initial definition of the interconnectivity vector proves to be unstable, but we introduce a stabilized version of the vector and prove its stability with respect to small perturbations in the inputs. We evaluate both versions of the presented vectorization on several data sets and show their high discriminative power.

cs.CG↗

arXiv2019

Detection of gravitational waves using topological data analysis and convolutional neural network: An improved approach

The gravitational wave detection problem is challenging because the noise is typically overwhelming. Convolutional neural networks (CNNs) have been successfully applied, but require a large training set and the accuracy suffers significantly in the case of low SNR. We propose an improved method that employs a feature extraction step using persistent homology. The resulting method is more resilient to noise, more capable of detecting signals with varied signatures and requires less training. This is a powerful improvement as the detection problem can be computationally intense and is concerned with a relatively large class of wave signatures.

astro-ph.IM↗

arXiv2016

Non-polynomial ENO and WENO finite volume methods for hyperbolic conservation laws

The essentially non-oscillatory (ENO) method is an efficient high order numerical method for solving hyperbolic conservation laws designed to reduce the Gibbs oscillations, if existent, by adaptively choosing the local stencil for the interpolation. The original ENO method is constructed based on the polynomial interpolation and the overall rate of convergence provided by the method is uniquely determined by the total number of interpolation points involved for the approximation. In this paper, we propose simple non-polynomial ENO and weighted ENO (WENO) finite volume methods in order to enhance the local accuracy and convergence. We first adopt the infinitely smooth radial basis functions (RBFs) for a non-polynomial interpolation. Particularly we use the multi-quadric and Gaussian RBFs. The non-polynomial interpolation such as the RBF interpolation offers the flexibility to control the local error by optimizing the free parameter. Then we show that the non-polynomial interpolation can be represented as a perturbation of the polynomial interpolation. That is, it is not necessary to know the exact form of the non-polynomial basis for the interpolation. In this paper, we formulate the ENO and WENO methods based on the non-polynomial interpolation and derive the optimization condition of the perturbation. To guarantee the essentially non-oscillatory property, we switch the non-polynomial reconstruction to the polynomial reconstruction adaptively near the non-smooth area by using the monotone polynomial interpolation method. The numerical results show that the developed non-polynomial ENO and WENO methods enhance the local accuracy.

math.NA↗

arXiv2016

Radial basis function ENO and WENO finite difference methods based on the optimization of shape parameters

We present adaptive finite difference ENO/WENO methods by adopting infinitely smooth radial basis functions (RBFs). This is a direct extension of the non-polynomial finite volume ENO/WENO method proposed by authors in \cite{GuoJung} to the finite difference ENO/WENO method based on the original smoothness indicator scheme developed by Jiang and Shu \cite{WENO}. The RBF-ENO/WENO finite difference method slightly perturbs the reconstruction coefficients with RBFs as the reconstruction basis and enhances accuracy in the smooth region by locally optimizing the shape parameters. The RBF-ENO/WENO finite difference methods provide more accurate reconstruction than the regular ENO/WENO reconstruction and provide sharper solution profiles near the jump discontinuity. Furthermore the RBF-ENO/WENO methods are easy to implement in the existing regular ENO/WENO code. The numerical results in 1D and 2D presented in this work show that the proposed RBF-ENO/WENO finite difference method better performs than the regular ENO/WENO method.

math.NA↗