Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry

Abstract

Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker–Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker–Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated.

Effects of Stochastic Noises on Limit-Cycle Oscillations and Power Losses in Fusion Plasmas and Information Geometry

“…It will be of interest to further extend our analysis to other nonlinear oscillators, including more than one oscillator, to quantify the correlation among different oscillators or emergent phenomena, such as synchronisation. This will require solving the Fokker–Planck equation in higher dimensions, which will be computationally challenging, or further developing stochastic simulation methods for an accurate calculation of information geometry diagnostics [ 40 ]. Finally, it would be interesting to investigate more general ELM models.…”

“…To ensure the accurate calculation of time-dependent PDFs and information diagnostics, we will use the Fokker–Planck method [ 39 ] instead of stochastic simulations of Equations ( 1 )–( 3 ) [ 40 ]. Corresponding to the Langevin model in Equations ( 1 )–( 3 ), the Fokker–Planck equation [ 39 ] for the joint PDF is given by …”

Non-equilibrium statistical properties, path-dependent information geometry, and entropy relations in edge-localized modes in fusion plasmas

“…An alternative method of direct stochastic simulations (e.g. in [65]) will be further developed to investigate these higherdimensional models. It will also be interesting to explore other information theoretical tools such as transfer entropy (e.g.…”

“…We note that an alternative method to the Fokker-Planck method is to perform multiple, stochastic simulations of Eqs. ( 2)-( 4) [65]. This would however require dealing with various issues in processing noisy data to calculate time-dependent PDFs and information diagnostics accurately, which is circumvented in the Fokker-Planck method.…”

A stochastic model of edge-localized modes in magnetically confined plasmas

“…However, the computational cost increases significantly as the number of variables increases. On the other hand, the alternative method of stochastic simulations requires many multiple runs for high-quality time-dependent PDFs and also needs to overcome technical challenges in data binning for constructing PDFs and the accurate calculation of information diagnostics, as discussed in [54].…”