Figure - available from: Nonlinear Dynamics
This content is subject to copyright. Terms and conditions apply.
Source publication
The concept of quasi-potential plays a central role in understanding the mechanisms of rare events and characterizing the statistics of transition behaviors in stochastic dynamics. Despite its significance, the computation of quasi-potential is a challenging problem with limited existing techniques. In this paper, we devise a machine learning metho...
Similar publications
We review the results obtained in Gianfelice et al. (Commun Math Phys 313:745–779, 2012) and Gianfelice and Vaienti (J Stat Phys 181(1):163–211, 2020) on the stochastic and statistical stability of the classical Lorenz flow, where, looking at the Lorenz’63 ODE system as a simple—yet non trivial—model of the atmospheric circulation, the perturbation...
Citations
... For example, machine learning methods have been used to discover stochastic dynamical systems from sample path data via non-local Kramers-Moyal formulas, [21,22] physics-informed neural networks [23,24] and variational inference. [25] They have greatly improved the efficiency and accuracy of solving physical quantities of stochastic dynamical systems by computing the most probable path, [26,27] quasipotential [28][29][30] and probability density. [31] These experimental results have brought about potential applications of a combination of machine learning and stochastic dynamics, including forward and inverse problems. ...
In this paper, we present large deviation theory that characterizes the exponential estimate for rare events of stochastic dynamical systems in the limit of weak noise. We aim to consider next-to-leading-order approximation for more accurate calculation of mean exit time via computing large deviation prefactors with the research efforts 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 vector field. We corroborate the higher effectiveness and accuracy of our algorithm with two toy models. Numerical experiments demonstrate its powerful functionality in exploring internal mechanism of rare events triggered by weak random fluctuations.
... Prior information is another basis for model identification. Neural networks methods can carefully design input layers and loss functions to learn certain characteristics [25,26] or the SDE [7,10,27] without prior information. When the physical information is introduced into the neural network [7,10], better prediction can be achieved. ...
... where f , h and 2D can be replaced by the results obtained by the first-order accuracy framework. The noise at each step can be estimated by formula (25). The identified SDE is written by formula (10). ...
This paper presents a high-order accuracy data-driven identification framework for stochastic differential equation (SDE) with the Gaussian white noise. The framework improves the accuracy in discovering SDE from the data collected at low frequencies. It first approximates the increment of data as a one-step mapping. The truncated terms of the approximation are the higher order terms of the time step of the collected data. Therefore, the mapping can approximate the data collected with large time steps, i.e., low frequencies with high-order accuracy. Then, the framework approximates the mapping as the Fourier series with a high rate of convergence, which is called the double-even extended series. In this way, we can approximate the average change of SDE with high accuracy and do not need knowledge about the change. After determining the coefficients of the Fourier series from data, the new framework finds an accurate approximation of the SDE. We apply the new framework to achieve the identification of an SD oscillator and a glycolytic oscillation. The new framework accurately discovers the SDEs from data collected at low frequencies, e.g., 4 HZ and 2.5 HZ. Identified SDEs can accurately predict stochastic responses. To the best of the authors' knowledge, the frequency is much lower than existing SDE identification methods.
... Through viewing noise induced transition event as an optimal con-trol problem, Wei et al. developed a neural network architechture to calculate the most probable paths of the stochastic systems with jumps [27]. There are also some proposed machine learning methods devoted to the computation of quasipotential based on orthogonal decomposition of the vector field or Hamilton-Jacobi equation [28,29]. ...
... The proposed deep learning method is actually a better alternative to the shooting method, retaining the advantage of its simple algorithm and removing the disadvantage of difficulty in adjusting initial values. In addition, we developed another machine learning method for computing quasipotential based on solving Hamilton-Jacobi equation recently [28]. Comparing with it, the method in this paper has the advantage that it can compute not only the quasipotential but also the prefactor of the saddle so that the mean exit time can be approximated more accurately. ...
... Comparing with it, the method in this paper has the advantage that it can compute not only the quasipotential but also the prefactor of the saddle so that the mean exit time can be approximated more accurately. However, it also has the limitation that we can only obtain the value of one point without global quasipotential landscape, which is the superiority of the previous method [28]. Which algorithm to choose depends on the kind of problem we are interested in. ...
Exit events induced by noise from the attracting domain containing a stable fixed point are ubiquitous phenomena in physical systems, wherein mean exit time is an important quantity which has been widely used in engineering, physical, chemical and biological fields. In this work, we devise a deep learning method to compute the mean exit time for dynamical systems excited by weak Gaussian noise. More specifically, we first derive a complete group of ordinary differential equations governing the most probable path, the quasipotential, and the prefactor along the path via WKB approximation. Then a neural network architechture is proposed to solve it in terms of automatic differentiation. The results of numerical experiments show the effectiveness and accuracy of the algorithm and imply its potential applications to discover the mechanisms of rare events triggered by random fluctuations in practical models.
... Discovering governing equations for complex problems such as climate, life, mechanics, finance, and gene is the key to revealing the nature of systems and predicting their motion [1][2][3][4][5][6][7][8]. A moving object is often motivated by time-dependent factors [9][10][11]. ...
A moving object is often excited by time-dependent factors. If the equation of motion and time-dependent factor can be distilled from data, a foundation for further analysis can be established. This paper proposes a method to discover the equation of motion and time-dependent factors without prior information and customized preparation. We first construct statistics to determine whether the system is subject to time-dependent factors. For the motivated system, we introduce the time-dependent Fourier series to approximate the excitation. Finally, we achieve the identification by simple steps. The new method does not require the prior knowledge of the system. Since the new method determines whether the time-dependent factor is present and which state variable is motivated, it avoids the confusion caused by inappropriately adding time-dependent terms to the identification. As the designed Fourier series can approximate the unknown model accurately, the new method does not require developing customized identification for each problem. We apply the new method to discover dynamical systems from data collected from Liénard equation and SD oscillator driven by various time-dependent factors. The new method can accurately discover dynamical systems from the collected short-time data and predict many dynamical behaviors, including the long-time evolution of the system, the multi-stable dynamics, and qualitative changes in the dynamics as the time-dependent factor varies. The results show that the new identification method can discover nonautonomous dynamical systems effectively.
... In the fields of stochastic dynamics, some data-driven methods have been established to successfully extract the non-Gaussian governing laws only from mean exit time [27] and extract the stochastic differential equation only from system state time series [28]. In addition, the quasipotential, which is quite important to describe the statistics of transition behaviors, is calculated by a machine learning method [29]. All of these validate the tremendous potential of the data-driven methodology to stochastic dynamics, whether in system identification or in behavior analysis. ...
For stochastically excited dissipative dynamical systems, the low-dimensional slowly varying processes act as the essential and simplified description of the apparent high-dimensional fast-varying processes (i.e., state variables). Deriving the statistical information of low-dimensional processes has a great significance, which reflects almost all the statistical information of concern. This work is devoted to an equation-free, data-driven method, which starts from random state data, automatically extracts the slowly varying processes and automatically identifies its stationary probability density. The independent slowly varying processes are extracted by combining the identification of Lagrangian and Legendre transformations; the probability density is identified by the assumption of exponential form and the comparison with calculated data at lattices; both steps are implemented in the framework of linear regression. This method is universally valid for general nonlinear systems with arbitrary parameter values; for systems with heavy damping and/or strong excitations, it provides sparse results with high precision, while the results from stochastic averaging are incorrect even in function property.
... Reservoir computing (RC), as a new type of recurrent neural network, was originally proposed by Wolfgang Maass in 2002. [16][17][18][19][20][21][22][23][24][25][26][27][28] At present, reservoir computing has shown very good properties in the processing of time series data. However, in the existing research studies that use deep learning algorithms to solve control problems, no scholars have used reservoir computing; most scholars still use traditional methods to solve the control prediction problem of chaotic systems. ...
Chaos is an important dynamic feature, which generally occurs in deterministic and stochastic nonlinear systems and is an inherent characteristic that is ubiquitous. Many difficulties have been solved and new research perspectives have been provided in many fields. The control of chaos is another problem that has been studied. In recent years, a recurrent neural network has emerged, which is widely used to solve many problems in nonlinear dynamics and has fast and accurate computational speed. In this paper, we employ reservoir computing to control chaos in dynamic systems. The results show that the reservoir calculation algorithm with a control term can control the chaotic phenomenon in a dynamic system. Meanwhile, the method is applicable to dynamic systems with random noise. In addition, we investigate the problem of different values for neurons and leakage rates in the algorithm. The findings indicate that the performance of machine learning techniques can be improved by appropriately constructing neural networks.
... Self-learning resulting from experience can occur within networks, which can derive conclusions from a complex and seemingly unrelated set of information [24]. It is worth pointing out that machine learning provides a means of tackling nondeterministic systems, where the equations used to model the system are not known [16]. ...
... According to the properties of quasipotential, the post-training quasipotential V(x) and momentum p(x) need to satisfy three constraints: p = ∇V(x), Hamilton-Jacobi equation, and the quasipotential of the stable fixed point x s being zero. We select N H collocation points on phase space to realize these constraints in the loss function [16]. ...
The mean exit time escaping basin of attraction in the presence of white noise is of practical importance in various scientific fields. In this work, we propose a strategy to control mean exit time of general stochastic dynamical systems to achieve a desired value based on the quasipotential concept and machine learning. Specifically, we develop a neural network architecture to compute the global quasipotential function. Then we design a systematic iterated numerical algorithm to calculate the controller for a given mean exit time. Moreover, we identify the most probable path between metastable attractors with help of the effective Hamilton-Jacobi scheme and the trained neural network. Numercal experiments demonstrate that our control strategy is effective and sufficiently accurate.
