arXiv subjects

Kayo Kinjo

Publications and source records attributed to Kayo Kinjo.

7 recordsLinked to original sources

arXiv2025

Permutation of Tensor-Train Cores for Computing Moments on Stochastic Differential Equations

Tensor networks, particularly the tensor train (TT) format, have emerged as powerful tools for high-dimensional computations in physics and computer science. In solving coupled differential equations, such as those arising from stochastic differential equations (SDEs) via duality relations, ordering the TT cores significantly influences numerical accuracy. In this study, we first systematically investigate how different orderings of the TT cores affect the accuracy of computed moments using the duality relation in stochastic processes. Through numerical experiments on a two-body interaction model, we demonstrate that specific orderings of the TT cores yield lower relative errors, particularly when they align with the underlying interaction structure of the system. Motivated by these findings, we then propose a novel quantitative measure, $score$, which is defined based on an ordering of the TT cores and an SDE parameter set. While the score is independent of the accuracy of moments to compute by definition, we assess its effectiveness by evaluating the accuracy of computed moments. Our results indicate that orderings that minimize the score tend to yield higher accuracy. This study provides insights into optimizing orderings of the TT cores, which is essential for efficient and reliable high-dimensional simulations of stochastic processes.

physics.comp-ph↗

arXiv2025

Integrated utilization of equations and small dataset in the Koopman operator: applications to forward and inverse problems

In recent years, there has been a growing interest in data-driven approaches in physics, such as extended dynamic mode decomposition (EDMD). The EDMD algorithm focuses on nonlinear time-evolution systems, and the constructed Koopman matrix yields the next-time prediction with only linear matrix-product operations. Note that data-driven approaches generally require a large dataset. However, assume that one has some prior knowledge, even if it may be ambiguous. Then, one could achieve sufficient learning from only a small dataset by taking advantage of the prior knowledge. This paper yields methods for incorporating ambiguous prior knowledge into the EDMD algorithm. The ambiguous prior knowledge in this paper corresponds to the underlying time-evolution equations with unknown parameters. First, we apply the proposed method to forward problems, i.e., prediction tasks. Second, we propose a scheme to apply the proposed method to inverse problems, i.e., parameter estimation tasks. We demonstrate the learning with only a small dataset using guiding examples, i.e., the Duffing and the van der Pol systems.

cs.LG↗

arXiv2024

Improvement of system identification of stochastic systems via Koopman generator and locally weighted expectation

The estimation of equations from data is of interest in physics. One of the famous methods is the sparse identification of nonlinear dynamics (SINDy), which utilizes sparse estimation techniques to estimate equations from data. Recently, a method based on the Koopman operator has been developed; the generator extended dynamic mode decomposition (gEDMD) estimates a time evolution generator of dynamical and stochastic systems. However, a naive application of the gEDMD algorithm cannot work well for stochastic differential equations because of the noise effects in the data. Hence, the estimation based on conditional expectation values, in which we approximate the first and second derivatives on each coordinate, is practical. A naive approach is the usage of locally weighted expectations. We show that the naive locally weighted expectation is insufficient because of the nonlinear behavior of the underlying system. For improvement, we apply the clustering method in two ways; one is to reduce the effective number of data, and the other is to capture local information more accurately. We demonstrate the improvement of the proposed method for the double-well potential system with state-dependent noise.

math.DS↗

arXiv2024

Extraction of nonlinearity in neural networks with Koopman operator

Nonlinearity plays a crucial role in deep neural networks. In this paper, we investigate the degree to which the nonlinearity of the neural network is essential. For this purpose, we employ the Koopman operator, extended dynamic mode decomposition, and the tensor-train format. The Koopman operator approach has been recently developed in physics and nonlinear sciences; the Koopman operator deals with the time evolution in the observable space instead of the state space. Since we can replace the nonlinearity in the state space with the linearity in the observable space, it is a hopeful candidate for understanding complex behavior in nonlinear systems. Here, we analyze learned neural networks for the classification problems. As a result, the replacement of the nonlinear middle layers with the Koopman matrix yields enough accuracy in numerical experiments. In addition, we confirm that the pruning of the Koopman matrix gives sufficient accuracy even at high compression ratios. These results indicate the possibility of extracting some features in the neural networks with the Koopman operator approach.

cs.LG↗

arXiv2023

Comparison of gap-based and flow-based control strategies using a new controlled stochastic cellular automaton model for traffic flow

Autonomous vehicles are essential to future transportation systems, potentially reducing traffic congestion. This study examines the impact of different vehicle control strategies on traffic flow through simulations. We propose a novel stochastic cellular automaton model, the controlled stochastic optimal velocity (CSOV) model, which incorporates vehicle control effects. Within the CSOV model, two control strategies are implemented: gap-based control (GC), which adjusts vehicle velocity to balance the gaps between adjacent vehicles, and flow-based control (FC), which aims to maintain a consistent local flow between the front and rear vehicles. Results show that both control strategies improve traffic flow. However, under weaker control, the GC sometimes resulted in lower flow compared to no control. In contrast, the FC consistently enhanced flow across control strengths, yielding more robust outcomes. Furthermore, when both strategies achieved comparable flow rates, the FC provided a more stable velocity distribution under varying traffic densities than the GC.

nlin.CG↗

arXiv2022

Dynamics of quantum double dark-solitons and an exact finite-size scaling of Bose-Einstein condensation

We show several novel aspects in the exact non-equilibrium dynamics of quantum double dark-soliton states in the Lieb-Liniger model for the one-dimensional Bose gas with repulsive interactions. We also show an exact finite-size scaling of the fraction of the Bose-Einstein condensation (BEC) in the ground state, which should characterize the quasi-BEC in quantum double dark-soliton states that we assume to occur in the weak coupling regime. First, we show the exact time evolution of the density profile in the quantum state associated with a quantum double dark-soliton by the Bethe ansatz. Secondly, we derive a kind of macroscopic quantum wave-function effectively by exactly evaluating the square amplitude and phase profiles of the matrix element of the field operator between the quantum double dark-soliton states. The profiles are close to those of dark-solitons particularly in the weak-coupling regime. Then, the scattering of two notches in the quantum double dark-soliton state is exactly demonstrated. It is suggested from the above observations that the quasi-BEC should play a significant role in the dynamics of quantum double dark-soliton states. If the condensate fraction is close to 1, the quantum state should be well approximated by the quasi-BEC state where the mean-field picture is valid.

cond-mat.quant-gas↗

arXiv2021

Quantum dark solitons in the 1D Bose gas: From single to double dark-solitons

We study quantum double dark-solitons by constructing corresponding quantum states in the Lieb-Liniger model for the one-dimensional Bose gas. Here we expect that the Gross-Pitaevskii (GP) equation should play a central role in the long distance mean-field behavior of the 1D Bose gas. We first introduce novel quantum states of a single dark soliton with a nonzero winding number. We show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the $N$-particle and $(N-1)$-particle states. For elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions, respectively, in the weak coupling regime. We then show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. Assigning the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations we obtain double dark-solitons of distinct narrow notches with different depths. We suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons.

cond-mat.quant-gas↗