We present a large deviation theory that characterizes the exponential estimate for rare events in stochastic dynamical systems in the limit of weak noise.We aim to consider a next-to-leading-order approximation for m...We present a large deviation theory that characterizes the exponential estimate for rare events in stochastic dynamical systems in the limit of weak noise.We aim to consider a next-to-leading-order approximation for more accurate calculation of the mean exit time by computing large deviation prefactors with the aid of machine learning.More specifically,we design a neural network framework to compute quasipotential,most probable paths and prefactors based on the orthogonal decomposition of a vector field.We corroborate the higher effectiveness and accuracy of our algorithm with two toy models.Numerical experiments demonstrate its powerful functionality in exploring the internal mechanism of rare events triggered by weak random fluctuations.展开更多
The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macrosc...The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macroscopic variables in the stochastic dynamic system. Traditional methods for solving these equations often struggle with computational efficiency and scalability, particularly in high-dimensional contexts. To address these challenges, this paper proposes a novel deep learning method based on prior knowledge with dual training to solve the stationary FPK equations. Initially, the neural network is pre-trained through the prior knowledge obtained by Monte Carlo simulation(MCS). Subsequently, the second training phase incorporates the FPK differential operator into the loss function, while a supervisory term consisting of local maximum points is specifically included to mitigate the generation of zero solutions. This dual-training strategy not only expedites convergence but also enhances computational efficiency, making the method well-suited for high-dimensional systems. Numerical examples, including two different two-dimensional(2D), six-dimensional(6D), and eight-dimensional(8D) systems, are conducted to assess the efficacy of the proposed method. The results demonstrate robust performance in terms of both computational speed and accuracy for solving FPK equations in the first three systems. While the method is also applicable to high-dimensional systems, such as 8D, it should be noted that computational efficiency may be marginally compromised due to data volume constraints.展开更多
Hamilton energy,which reflects the energy variation of systems,is one of the crucial instruments used to analyze the characteristics of dynamical systems.Here we propose a method to deduce Hamilton energy based on the...Hamilton energy,which reflects the energy variation of systems,is one of the crucial instruments used to analyze the characteristics of dynamical systems.Here we propose a method to deduce Hamilton energy based on the existing systems.This derivation process consists of three steps:step 1,decomposing the vector field;step 2,solving the Hamilton energy function;and step 3,verifying uniqueness.In order to easily choose an appropriate decomposition method,we propose a classification criterion based on the form of system state variables,i.e.,type-I vector fields that can be directly decomposed and type-II vector fields decomposed via exterior differentiation.Moreover,exterior differentiation is used to represent the curl of low-high dimension vector fields in the process of decomposition.Finally,we exemplify the Hamilton energy function of six classical systems and analyze the relationship between Hamilton energy and dynamic behavior.This solution provides a new approach for deducing the Hamilton energy function,especially in high-dimensional systems.展开更多
基金Project supported by the Natural Science Foundation of Jiangsu Province (Grant No.BK20220917)the National Natural Science Foundation of China (Grant Nos.12001213 and 12302035)。
文摘We present a large deviation theory that characterizes the exponential estimate for rare events in stochastic dynamical systems in the limit of weak noise.We aim to consider a next-to-leading-order approximation for more accurate calculation of the mean exit time by computing large deviation prefactors with the aid of machine learning.More specifically,we design a neural network framework to compute quasipotential,most probable paths and prefactors based on the orthogonal decomposition of a vector field.We corroborate the higher effectiveness and accuracy of our algorithm with two toy models.Numerical experiments demonstrate its powerful functionality in exploring the internal mechanism of rare events triggered by weak random fluctuations.
基金Project supported by the National Natural Science Foundation of China (Grant No.12172226)。
文摘The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macroscopic variables in the stochastic dynamic system. Traditional methods for solving these equations often struggle with computational efficiency and scalability, particularly in high-dimensional contexts. To address these challenges, this paper proposes a novel deep learning method based on prior knowledge with dual training to solve the stationary FPK equations. Initially, the neural network is pre-trained through the prior knowledge obtained by Monte Carlo simulation(MCS). Subsequently, the second training phase incorporates the FPK differential operator into the loss function, while a supervisory term consisting of local maximum points is specifically included to mitigate the generation of zero solutions. This dual-training strategy not only expedites convergence but also enhances computational efficiency, making the method well-suited for high-dimensional systems. Numerical examples, including two different two-dimensional(2D), six-dimensional(6D), and eight-dimensional(8D) systems, are conducted to assess the efficacy of the proposed method. The results demonstrate robust performance in terms of both computational speed and accuracy for solving FPK equations in the first three systems. While the method is also applicable to high-dimensional systems, such as 8D, it should be noted that computational efficiency may be marginally compromised due to data volume constraints.
基金the National Natural Science Foundation of China(Grant Nos.12305054,12172340,and 12371506)。
文摘Hamilton energy,which reflects the energy variation of systems,is one of the crucial instruments used to analyze the characteristics of dynamical systems.Here we propose a method to deduce Hamilton energy based on the existing systems.This derivation process consists of three steps:step 1,decomposing the vector field;step 2,solving the Hamilton energy function;and step 3,verifying uniqueness.In order to easily choose an appropriate decomposition method,we propose a classification criterion based on the form of system state variables,i.e.,type-I vector fields that can be directly decomposed and type-II vector fields decomposed via exterior differentiation.Moreover,exterior differentiation is used to represent the curl of low-high dimension vector fields in the process of decomposition.Finally,we exemplify the Hamilton energy function of six classical systems and analyze the relationship between Hamilton energy and dynamic behavior.This solution provides a new approach for deducing the Hamilton energy function,especially in high-dimensional systems.
