• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.28 no.1
• /
• pp.33-58
• /
• 2024

This paper reviews basic concepts for Physics-Informed Neural Networks (PINN) applied to the initial value problems for ordinary differential equations. In particular, using only basic calculus, we derive the error estimates where the error functions (the differences between the true solution and the approximations expressed by neural networks) are dominated by training loss functions. Numerical experiments are conducted to validate our error estimates, visualizing the relationship between the error and the training loss for various first-order differential equations and a second-order linear equation.

• Korean Journal of Mathematics
• /
• v.28 no.2
• /
• pp.223-240
• /
• 2020

The paper is concerned with periodic linear evolution equations of the form x"(t) = A(t)x(t)+f(t), where A(t) is a family of (unbounded) linear operators in a Banach space X, strongly and periodically depending on t, f is an almost (or asymptotic) almost periodic function. We study conditions for this equation to have almost periodic solutions on ℝ as well as to have asymptotic almost periodic solutions on ℝ⁺. We convert the second order equation under consideration into a first order equation to use the spectral theory of functions as well as recent methods of study. We obtain new conditions that are stated in terms of the spectrum of the monodromy operator associated with the first order equation and the frequencies of the forcing term f.

• Korean Journal of Mathematics
• /
• v.24 no.3
• /
• pp.573-586
• /
• 2016

We study the conditional integral transforms and conditional convolutions of functionals defined on K[0, T]. We consider a general vector-valued conditioning functions $X_k(x)=({\gamma}_1(x),{\ldots},{\gamma}_k(x))$ where ${\gamma}_j(x)$ are Gaussian random variables on the Wiener space which need not depend upon the values of x at only finitely many points in (0, T]. We then obtain several relationships and formulas for the conditioning functions that exist among conditional integral transform, conditional convolution and first variation of functionals in $E_{\sigma}$.

• Journal of the Korean Society of Earth Science Education
• /
• v.10 no.2
• /
• pp.214-225
• /
• 2017

The first accurate estimate of the Earth's circumference was made by the Hellenism scientist Eratosthenes (276-195 B.C.) in about 240 B.C. The simplicity and elegance of Eratosthenes' measurement of the circumference of the Earth by mathematics abstraction strategies were an excellent example of ancient Greek ingenuity. Eratosthenes's success was a triumph of logic and the scientific method, the method required that he assume that Sun was so far away that its light reached Earth along parallel lines. That assumption, however, should be supported by another set of measurements made by the ancient Hellenism, Aristarchus, namely, a rough measurement of the relative diameters and distances of the Sun and Moon. Eratosthenes formulated the simple proportional formula, by mathematic abstraction strategies based on perfect sphere and a simple mathematical rule as well as in the geometry in this world. The Earth must be a sphere by a logical and empirical argument of Aristotle, based on the Greek word symmetry including harmony and beauty of form. We discuss the justification of these three bold assumptions for mathematical abstraction of Eratosthenes's experiment for calculating the circumference of the Earth, and justifying all three assumptions from historical perspective for mathematics and science education. Also it is important that the simplicity about the measurement of the earth's circumstance at the history of science.

• KSII Transactions on Internet and Information Systems (TIIS)
• /
• v.14 no.1
• /
• pp.168-182
• /
• 2020

Spoofing attacks, especially replay attacks, pose great security challenges to automatic speaker verification (ASV) systems. Current works on replay attacks detection primarily focused on either developing new features or improving classifier performance, ignoring the effects of feature variability, e.g., the channel variability. In this paper, we first establish a mathematical model for replay speech and introduce a method for eliminating the negative interference of the channel. Then a novel feature is proposed to detect the replay attacks. To further boost the detection performance, four post-processing methods using normalization techniques are investigated. We evaluate our proposed method on the ASVspoof 2017 dataset. The experimental results show that our approach outperforms the competing methods in terms of detection accuracy. More interestingly, we find that the proposed normalization strategy could also improve the performance of the existing algorithms.

• Bulletin of the Korean Chemical Society
• /
• v.35 no.4
• /
• pp.1057-1060
• /
• 2014

In our previous articles, an approach has been proposed for the evaluation of the uncertainty of overall result from multiple measurements. In the approach, uncertainty sources were classified into two groups: the first including those giving same 'systematic' effect on each individual measurement and the second including the others giving 'random' effect on each individual measurement and causing a variation among individual measurement results. The arithmetic mean of the replicated measurements is usually assigned as the value for the overall result. Uncertainty of the overall result is determined by separately evaluating and combining an overall uncertainty from sources of the 'systematic' effect and another overall uncertainty from sources of the 'random' effect. This conceptual approach has been widely adopted in chemical metrology society. In this study, further logical proof with more detailed mathematical expressions is provided on the approach.

• Composites Research
• /
• v.15 no.5
• /
• pp.44-52
• /
• 2002

To study numerically the mechanical behaviors of advanced composite materials considering the microscopic phenomena as well as the macroscopic properties and behaviors, a multi-scale modeling and analysis by the mathematical homogenization method with the help of the finite element method(FEM) are reviewed. The hierarchical modeling strategy and the formulation are briefly described first to give some idea of the multi-scale framework. The latter half of this article focuses on the verification of the multi-scale analysis by the homogenization method in its applications to real advanced materials. The first example is the verification of the predicted macroscopic(homogenized) properties based on the microstructure of porous ceramics. In spite of the complexity of the random microstructure, the error between the predicted and the measured values was only 1%. Next, two applications to the process simulation of fiber reinforced polymer matrix composites are presented. The permeability characteristics are evaluated for sheared weave fabrics for resin transfer molding(RTM) simulation, and the thermoforming of FRTP sheet is analyzed considering the large deformation of the knit structure during the deep-draw forming was verified by comparison with the experimental results.

• Journal of Computing Science and Engineering
• /
• v.7 no.2
• /
• pp.99-111
• /
• 2013

Approximating a divergence between two probability distributions from their samples is a fundamental challenge in statistics, information theory, and machine learning. A divergence approximator can be used for various purposes, such as two-sample homogeneity testing, change-point detection, and class-balance estimation. Furthermore, an approximator of a divergence between the joint distribution and the product of marginals can be used for independence testing, which has a wide range of applications, including feature selection and extraction, clustering, object matching, independent component analysis, and causal direction estimation. In this paper, we review recent advances in divergence approximation. Our emphasis is that directly approximating the divergence without estimating probability distributions is more sensible than a naive two-step approach of first estimating probability distributions and then approximating the divergence. Furthermore, despite the overwhelming popularity of the Kullback-Leibler divergence as a divergence measure, we argue that alternatives such as the Pearson divergence, the relative Pearson divergence, and the $L^2$-distance are more useful in practice because of their computationally efficient approximability, high numerical stability, and superior robustness against outliers.

• Journal of the Korean Chemical Society
• /
• v.64 no.2
• /
• pp.119-129
• /
• 2020

For effective teaching-learning activities for students with diverse talents in science high schools, it is important for teachers to understand students' individual differences in perceiving and processing information in the natural world, depending on the students' various talents and subject characteristics. The purpose of this study is to examine the students' cognition of chemistry in science high school through correlations and factor analysis of mathematics/science achievement. In addition, this study attempted to examine the cognition of chemistry subject according to R&E classes. The main participants of the study were freshmen of G science high school (296 students) who entered after three times of curriculum reforms and new admission processes and the students in two other science high schools in Gyeongnam and Ulsan were included. The correlation and factor analysis were conducted by exploratory factor analysis by IBM SPSS Statistics 25 programs. The results of this study were as follows: First, in the correlation analysis between mathematics and science achievement, it was confirmed that the Pearson's coefficient of chemistry showed higher positive correlation coefficient than that of other science subjects. Second, in the factor analysis of mathematics and science achievements, it was found that the factor indicators were divided into two factors as logical-mathematical (mathematics and physics) and naturalistic (life science and earth science). Third, in the factor analysis, it was confirmed that the chemistry is recognized as the subject that requires both logical-mathematical and naturalistic intelligence. Finally, it was confirmed that students' cognitions of chemistry subject were found to differ according to the R&E classes. In other words, the participants of R&E chemistry class, unlike other students, were found to recognize chemistry as the subject that logical-mathematical intelligence is needed.

• Korean Journal of Cognitive Science
• /
• v.23 no.3
• /
• pp.367-388
• /
• 2012

The centennial of Alan Mathison Turing(23 June 1912 - 7 June 1954) is an appropriate occasion on which to assess his profound influence on the development of cognitive science. His contributions to and attitudes toward that field are discussed from the metamathematical perspective. This essay addresses (i)Turing's mathematical analysis of cognition, (ii)universal Turing machines, (iii)the limitations of universal Turing machines, (iv)oracle Turing machine beyond universal Turing machine, and (v)Turing test for cognitive science. Turing was a ground-breaker, eager to move on to new fields. He actually opened wider the scientific windows to the mind. The results show that first, by means of mathematical logic Turing discovered a new bridge between the mind and the physical world. Second, Turing gave a new formal analysis of operations of the mind. Third, Turing investigated oracle Turing machines and connectionist network machines as new models of minds beyond the limitations of his own universal machines. This paper explores why the cognitive scientist would be ever expecting a new Turing Test on the shoulder of Alan Turing.