arXiv subjects

Jun Ohkubo

Publications and source records attributed to Jun Ohkubo.

At least 37 records · Page 2Linked to original sources

arXiv2023

Embedding stochastic differential equations into neural networks via dual processes

We propose a new approach to constructing a neural network for predicting expectations of stochastic differential equations. The proposed method does not need data sets of inputs and outputs; instead, the information obtained from the time-evolution equations, i.e., the corresponding dual process, is directly compared with the weights in the neural network. As a demonstration, we construct neural networks for the Ornstein-Uhlenbeck process and the noisy van der Pol system. The remarkable feature of learned networks with the proposed method is the accuracy of inputs near the origin. Hence, it would be possible to avoid the overfitting problem because the learned network does not depend on training data sets.

cs.LG↗

arXiv2023

Statistics for stochastic differential equations and approximations of resolvent

The numerical evaluation of statistics plays a crucial role in statistical physics and its applied fields. It is possible to evaluate the statistics for a stochastic differential equation with Gaussian white noise via the corresponding backward Kolmogorov equation. The important notice is that there is no need to obtain the solution of the backward Kolmogorov equation on the whole domain; it is enough to evaluate a value of the solution at a certain point that corresponds to the initial coordinate for the stochastic differential equation. For this aim, an algorithm based on combinatorics has recently been developed. In this paper, we discuss a higher-order approximation of resolvent, and an algorithm based on a second-order approximation is proposed. The proposed algorithm shows a second-order convergence. Furthermore, the convergence property of the naive algorithms naturally leads to extrapolation methods; they work well to calculate a more accurate value with fewer computational costs. The proposed method is demonstrated with the Ornstein-Uhlenbeck process and the noisy van der Pol system.

math.NA↗

arXiv2023

Efficient correlation-based discretization of continuous variables for annealing machines

Annealing machines specialized for combinatorial optimization problems have been developed, and some companies offer services to use those machines. Such specialized machines can only handle binary variables, and their input format is the quadratic unconstrained binary optimization (QUBO) formulation. Therefore, discretization is necessary to solve problems with continuous variables. However, there is a severe constraint on the number of binary variables with such machines. Although the simple binary expansion in the previous research requires many binary variables, we need to reduce the number of such variables in the QUBO formulation due to the constraint. We propose a discretization method that involves using correlations of continuous variables. We numerically show that the proposed method reduces the number of necessary binary variables in the QUBO formulation without a significant loss in prediction accuracy.

quant-ph↗

arXiv2022

Numerical methods to evaluate Koopman matrix from system equations

The Koopman operator is beneficial for analyzing nonlinear and stochastic dynamics; it is linear but infinite-dimensional, and it governs the evolution of observables. The extended dynamic mode decomposition (EDMD) is one of the famous methods in the Koopman operator approach. The EDMD employs a data set of snapshot pairs and a specific dictionary to evaluate an approximation for the Koopman operator, i.e., the Koopman matrix. In this study, we focus on stochastic differential equations, and a method to obtain the Koopman matrix is proposed. The proposed method does not need any data set, which employs the original system equations to evaluate some of the targeted elements of the Koopman matrix. The proposed method comprises combinatorics, an approximation of the resolvent, and extrapolations. Comparisons with the EDMD are performed for a noisy van der Pol system. The proposed method yields reasonable results even in cases wherein the EDMD exhibits a slow convergence behavior.

math.NA↗

arXiv2021

Connection among stochastic Hamilton-Jacobi-Bellman equation, path-integral, and Koopman operator on nonlinear stochastic optimal control

The path-integral control, which stems from the stochastic Hamilton-Jacobi-Bellman equation, is one of the methods to control stochastic nonlinear systems. This paper gives a new insight into nonlinear stochastic optimal control problems from the perspective of Koopman operators. When a finite-dimensional dynamical system is nonlinear, the corresponding Koopman operator is linear. Although the Koopman operator is infinite-dimensional, adequate approximation makes it tractable and useful in some discussions and applications. Employing the Koopman operator perspective, it is clarified that only a specific type of observable is enough to be focused on in the control problem. This fact becomes easier to understand via path-integral control. Furthermore, the focus on the specific observable leads to a natural power-series expansion; coupled ordinary differential equations for discrete-state space systems are derived. A demonstration for nonlinear stochastic optimal control shows that the derived equations work well.

math.OC↗

arXiv2021

Semi-supervised learning combining backpropagation and STDP: STDP enhances learning by backpropagation with a small amount of labeled data in a spiking neural network

A semi-supervised learning method for spiking neural networks is proposed. The proposed method consists of supervised learning by backpropagation and subsequent unsupervised learning by spike-timing-dependent plasticity (STDP), which is a biologically plausible learning rule. Numerical experiments show that the proposed method improves the accuracy without additional labeling when a small amount of labeled data is used. This feature has not been achieved by existing semi-supervised learning methods of discriminative models. It is possible to implement the proposed learning method for event-driven systems. Hence, it would be highly efficient in real-time problems if it were implemented on neuromorphic hardware. The results suggest that STDP plays an important role other than self-organization when applied after supervised learning, which differs from the previous method of using STDP as pre-training interpreted as self-organization.

cs.NE↗

arXiv2020

Combinatorics for calculating expectation values of functions in systems with evolution governed by stochastic differential equations

Stochastic differential equations are widely used in various fields; in particular, the usefulness of duality relations has been demonstrated in some models such as population models and Brownian momentum processes. In this study, a discussion based on combinatorics is made and applied to calculate the expectation values of functions in systems in which evolution is governed by stochastic differential equations. Starting with the duality theory of stochastic processes, some modifications to the interpretation and usage of time-ordering operators naturally lead to discussions on combinatorics. For demonstration, the first and second moments of the Ornstein-Uhlenbeck process are re-derived from the discussion on combinatorics. Furthermore, two numerical methods for practical applications are proposed. One method is based on a conventional exponential expansion and the Pade approximation. The other uses a resolvent of a time-evolution operator, along with the application of the Aitken series acceleration method. Both methods yield reasonable approximations. Particularly, the resolvent and Aitken acceleration show satisfactory results. These findings will provide a new way of calculating expectations numerically and directly without using time-discretization.

cond-mat.stat-mech↗

arXiv2020

Making Birth-Death Processes from Backward Fokker-Planck Equations for Computing Expectations in Langevin Systems

A method to direct evaluation of expectations for Langevin systems (stochastic differential equations) is proposed. The method is based on a birth-death process which is derived using combinations of dummy variables and It{ô} formula. As a pedagogical example, a double-well system and expectations for sigmoid-type functions are used. It is shown that the proposed method has some merits from computational point of view; only one time-integration for the birth-death process gives expectations for various initial conditions in the original Langevin systems. Furthermore, the same time-integration result is available for computing various center positions of the sigmoid-type functions.

physics.comp-ph↗

arXiv2020

Derivation of QUBO formulations for sparse estimation

We propose a quadratic unconstrained binary optimization (QUBO) formulation of the l1-norm, which enables us to perform sparse estimation of Ising-type annealing methods such as quantum annealing. The QUBO formulation is derived using the Legendre transformation and the Wolfe theorem, which have recently been employed to derive the QUBO formulations of ReLU-type functions. It is shown that a simple application of the derivation method to the l1-norm case results in a redundant variable. Finally a simplified QUBO formulation is obtained by removing the redundant variable.

quant-ph↗

arXiv2019

Duality in stochastic processes from the viewpoint of basis expansions

A new derivation method of duality relations in stochastic processes is proposed. The current focus is on the duality between stochastic differential equations and birth-death processes. Although previous derivation methods have been based on the viewpoint of time-evolution operators, the current derivation is based on basis expansions. In addition, only the tool needed for the derivation is the integration by parts, which is rather simple and understandable. The viewpoint of basis expansions enables us to obtain various dual stochastic processes. As a demonstration, dual processes based on Taylor-type and Hermite polynomials are derived.

cond-mat.stat-mech↗

arXiv2019

Quadratic unconstrained binary optimization formulation for rectified-linear-unit-type functions

We propose a quadratic unconstrained binary optimization (QUBO) formulation of rectified linear unit (ReLU) type functions. Different from the q-loss function proposed by Denchev et al. (2012), a simple discussion based on the Legendre duality is not sufficient to obtain the QUBO formulation of the ReLU-type functions. In addition to the Legendre duality, we employ the Wolfe duality, and the QUBO formulation of the ReLU-type is derived. The QUBO formulation is available in Ising-type annealing methods, including quantum annealing machines.

quant-ph↗

arXiv2017

Duality-based calculations for transition probabilities in stochastic chemical reactions

An idea for evaluating transition probabilities in chemical reaction systems is proposed, which is efficient for repeated calculations with various rate constants. The idea is based on duality relations; instead of direct time-evolutions of the original reaction system, the dual process is dealt with. Usually, if one changes rate constants of the original reaction system, the direct time-evolutions should be performed again, using the new rate constants. On the other hands, only one solution of an extended dual process can be re-used to calculate the transition probabilities for various rate constant cases. The idea is demonstrated in a parameter estimation problem for the Lotka-Volterra system.

cond-mat.stat-mech↗

arXiv2016

On dualities for SSEP and ASEP with open boundary conditions

Duality relations for simple exclusion processes with general open boundaries are discussed. It is shown that a combination of spin operators and bosonic operators enables us to have an unified discussion for the duality relations with the open boundaries. As for the symmetric simple exclusion process (SSEP), more general results than previous studies are obtained; it is clarified that not only the absorbing sites, but also additional sites, called copying sites, are needed for the boundaries in the dual process for the SSEP. The role of the copying sites is to conserve information about the particle states on the boundary sites. The similar discussions are applied to the asymmetric simple exclusion process (ASEP), in which the q-analogues are employed, and it is clarified that the ASEP with open boundaries has a complicated dual process on the boundaries.

cond-mat.stat-mech↗

arXiv2015

Nonlinear Kalman filter based on duality relations between continuous and discrete-state stochastic processes

A new application of duality relations of stochastic processes is demonstrated. Although conventional usages of the duality relations need analytical solutions for the dual processes, we here employ numerical solutions of the dual processes and investigate the usefulness. As a demonstration, estimation problems of hidden variables in stochastic differential equations are discussed. Employing algebraic probability theory, a little complicated birth-death process is derived from the stochastic differential equations, and an estimation method based on the ensemble Kalman filter is proposed. As a result, the possibility for making faster computational algorithms based on the duality concepts is shown.

eess.SY↗

arXiv2014

Lie algebraic discussions for time-inhomogeneous linear birth-death processes with immigration

Analytical solutions for time-inhomogeneous linear birth-death processes with immigration are derived. While time-inhomogeneous linear birth-death processes without immigration have been studied by using a generating function approach, the processes with immigration are here analyzed by Lie algebraic discussions. As a result, a restriction for time-inhomogeneity of the birth-death process is understood from the viewpoint of the finiteness of the dimensionality of the Lie algebra.

math-ph↗

arXiv2014

Karlin-McGregor-like formula in a simple time-inhomogeneous birth-death process

A novel approach is employed and developed to derive transition probabilities for a simple time-inhomogeneous birth-death process. Algebraic probability theory and Lie algebraic treatments make it easy to treat the time-inhomogeneous cases. As a result, an expression based on the Charlier polynomials is obtained, which can be considered as an extension of a famous Karlin-KcGregor representation for a time-homogeneous birth-death process.

math-ph↗

arXiv2013

Extended duality relations between birth-death processes and partial differential equations

Duality relations between continuous-state and discrete-state stochastic processes with continuous-time have already been studied and used in various research fields. We propose extended duality relations, which enable us to derive discrete-state stochastic processes from arbitrary diffusion-type partial differential equations. The derivation is based on the Doi-Peliti formalism and the algebraic probability theory, and it will be clarified that additional states for the discrete-state stochastic processes must be considered in some cases.

cond-mat.stat-mech↗

arXiv2013

Noncyclic geometric phase in counting statistics and its role as an excess contribution

We propose an application of fiber bundles to counting statistics. The framework of the fiber bundles gives a splitting of a cumulant generating function for current in a stochastic process, i.e., contributions from the dynamical phase and the geometric phase. We will show that the introduced noncyclic geometric phase is related to a kind of excess contributions, which have been investigated a lot in nonequilibrium physics. Using a specific nonequilibrium model, the characteristics of the noncyclic geometric phase are discussed; especially, we reveal differences between a geometric contribution for the entropy production and the `excess entropy production' which has been used to discuss the second law of steady state thermodynamics.

cond-mat.stat-mech↗