Fokker Planck Kolmogorov Research Articles

Higher-order stochastic averaging for investigating a vehicle suspension system with nonlinear damping and stiffness

The paper deals with a quarter-car model with nonlinear damping and stiffness under white noise base excitation using the higher-order stochastic averaging method for analyzing approximate responses. Recently, a novel higher-order averaging procedure has been developed to find analytically the first-, second-, and third-order stationary joint probability density functions (PDF) of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation, and it will be extended to the nonlinear quarter-car model. Accordingly, the mean-square responses such as the displacement, and velocity of the sprung mass can be obtained analytically. The influences of excitation intensity on the dynamic responses, as well as the variations of linear and nonlinear damping, are analyzed. A very satisfactory agreement is found between the accuracy of the solutions corresponding to higher-order stochastic averaging and that of Monte Carlo simulation.

Computational Methods for Stationary and Nonstationary Fokker–Planck–Kolmogorov Equations with Random Coefficients

In this paper, methodological approaches for distribution functions of nonlinear stochastic differential equations (SDEs) with uncertain parameters are elaborated. The stationary and nonstationary Fokker–Planck–Kolmogorov (FPK) equations associated with the considered random equations are investigated. A semi-analytical approach based on the extended exponential closure handling the parameters uncertainty effects is presented. A new numerical method coupling the polynomial chaos with the differential quadrature method (DQM) is elaborated for the stationary case. A compact matrix formulation is established allowing considering a large number of generalized polynomial chaos. For nonstationary random FPK equations, the polynomial chaos is coupled with the quadrature discretization. A symmetric discretization of the main Fokker–Planck operator using a tensor product of the Hilbert state space and the Hilbert probabilistic space has resulted. The obtained eigenvalues and random eigenfunctions are used for a spectral decomposition. The time solution is obtained by the spectral expansion as well as by the random Euler stochastic method. The accuracy, effectiveness and advantages of the developed procedure in analyzing the stationary and nonstationary probabilistic solutions of nonlinear stochastic dynamical systems with uncertain parameters excited by a Gaussian white noise are demonstrated.

Using reservoir computing to solve FPK equations for stochastic dynamical systems under Gaussian or Non-Gaussian excitation

This paper presents a new approach that uses the Reservoir Computing Algorithm to solve Fokker-Planck-Kolmogorov (FPK) equation excited by both Gaussian white noise and non-Gaussian noise. Unlike typical numerical methods, this methodology does not necessitate spatial reconstruction or numerical supplementation. The novelty of this paper lies in the modifications made to the conventional Reservoir Computing algorithm. We altered the approach for calculating values of the input weight matrix and incorporated autoregressive techniques in the reservoir layer. In addition, we applied data normalization to the training data before training the algorithm to avoid a zero solution. The efficacy of this approach was verified through multiple arithmetic examples, showcasing its practicality and efficiency in solving FPK equations. Moreover, the Reservoir Computing-FPK algorithm is capable of solving high-dimensional and fractional-order FPK equations with a smaller training set than earlier algorithms. Finally, we analyzed how values of the input weight matrix and regularization parameter affected the performance of the algorithm. The findings suggest that the careful selection of hyperparameters can greatly improve the performance of the Reservoir Computing algorithm.

Random Vibration Analysis of Double Deck Rocking Self-Centering Pier Under Seismic Excitation

This paper investigates nonlinear random vibration of double deck rocking self-centering pier structure subjected to seismic excitation. First of all, a novel type of double deck rocking self-centering pier that offers better resilience is designed. The stochastic dynamical model of the system is then established, in which the seismic excitation is viewed as Kanai–Tajimi filtered white noise and the self-centering restoring force is characterized by classical flag-shaped hysteretic model. Subsequently, the self-centering restoring force is decomposed into equivalent quasi-linear elastic and damping forces by adopting the harmonic balance (HB) scheme. Following this, the stochastic averaging (SA) technique is implemented to derive the averaged Itô equation. The steady-state response probability density with respect to the amplitudes is solved from the averaged Fokker–Planck–Kolmogorov (FPK) equation. Finally, the effects of system parameters on the stochastic response of the double deck rocking self-centering pier structure are performed, and validated by the Monte Carlo simulation (MCS). In addition, with the help of the Laplace transform, the transition function as well as conditional power spectral density are achieved, which are combined with the steady-state probability density to obtain the power spectral density response. This research will contribute to the optimal seismic design of double-deck rocking self-centering piers.

An efficient method for solving high-dimension stationary FPK equation of strongly nonlinear systems under additive and/or multiplicative white noise

Engineering structures may suffer from drastic nonlinear random vibrations in harsh environments. Random vibration has been extensively studied since 1960s, but is still an open problem for large-scale strongly nonlinear systems. In this paper, a random vibration analysis method based on the Neural Networks for large-scale strongly nonlinear systems under additive and/or multiplicative Gaussian white noise (GWN) excitations is proposed. In the proposed method, the high-dimensional steady-state Fokker–Planck-Kolmogorov (FPK) equation governing the state’s probability density function (PDF) is firstly reduced to low-dimensional FPK equation involving only the interested state variables, generally one or two dimensions. The equivalent drift coefficients (EDCs) and diffusion coefficients (EDFs) in the low-dimensional FPK equation are proven to be the conditional mean of the coefficients given the interested variables. Furthermore, it is shown that the conditional mean can be optimally estimated by regression. Subsequently, the EDCs and EDFs, as functions of the retained variables, are approximated by the semi-analytical Radial Basis Functions Neural Networks trained with samples generated by a few deterministic analyses. Finally, the Physics Informed Neural Network is employed to solve the reduced steady-state FPK equation, and the PDF of the system responses is obtained. Four typical examples under additive and/or multiplicative GWN excitations are used to examine the accuracy and efficiency of the proposed method by comparing its results with the exact solution (if available) or Monte Carlo simulations. The proposed method also exhibits greater accuracy than the globally-evolving-based generalized density evolution equation scheme, a similar method of its kind, especially for strongly nonlinear systems. Notably, even though steady-state systems are applied in this paper, there is no obstacle to extending the proposed framework to transient systems.

Approximate averaged derivative moments and stochastic response of high-DOF quasi-non-integrable Hamiltonian systems under Poisson white noises

Stochastic response prediction is one of the principal topics in diverse branches of engineering, such as aerospace engineering, mechanical engineering, and vehicle engineering. However, the challenge of addressing high-dimensional partial differential equations or evaluating high-dimensional domain integrals in determining averaged derivative moments has limited the research. This study proposes a method to predict the stochastic response of high-degree-of-freedom (DOF) quasi-non-integrable Hamiltonian systems under Poisson white noises. The original finite DOF system is reduced to a one-dimensional one governing the Hamiltonian by applying the stochastic averaging method, and the associated reduced generalized Fokker–Planck–Kolmogorov (GFPK) equation is derived. Then, the two-step generalized elliptical coordinate transformation is extended to evaluate averaged derivative moments that are the key to obtaining the stationary probability density function (SPDF) for probabilistic information prediction. Moreover, the perturbation method is introduced to address the high-order GFPK equation. Finally, an 8-DOF strongly nonlinear system under purely additive Poisson white noises is given as an example to highlight the accuracy of the proposed method. Results from the proposed method show good agreement with those from Monte Carlo simulations, which indicates that the proposed method can be applied to assess the response of high-DOF quasi-non-integrable Hamiltonian systems under Poisson white noises. In addition, if the potential energy of an arbitrary finite n-DOF system is the same form as that in Appendix B, pertinent high-dimensional domain integrals are converted into explicit expressions when n is even.

Stochastic Roll Dynamics of Smooth and Impacting Vessels in Random Waves

Ship instability is frequently associated with extreme roll motion. In this paper smooth and vibro-impact dynamics of vessels rolling under random wave excitations including nonlinearities is investigated. The vessel is represented by a single degree of freedom roll model. The random excitation due to waves and ice is modeled by a sum of random pulses with Poisson arrival rate. Stochastic P-bifurcation analysis is carried out by examining the variation with parameter of the joint probability density functions (jpdf) of the response computed numerically by the solution of the corresponding Fokker–Planck–Kolmogorov (FPK) equation. The finite element (FE), finite difference (FD), path integral (PI) and radial basis neural (RBN) network formulation methods are used in the solution of the FPK equation. The D-bifurcation analysis is analyzed by computing the largest Lyapunov exponent. The mean upcrossing rate is computed using Rice’s formula. The random wave excitation is also modeled as the response of a second order linear filter to white noise and the random response of the ship is investigated by solution of the corresponding FPK equation of the four dimensional Markov system by the FD method. Non-smooth coordinate transformations like the Zhuravlev and Ivanov transformations are used converting the discontinuous systems to equivalent smooth systems in the impact dynamics analysis. The adaptive time stepping method (ATSM) approach is used to accurately locate the discontinuity point and a Brownian tree approach is used to direct the integration along the correct Brownian path. Different models for nonlinear damping and the restoring moment are considered and their effects on ship stochastic response such as chaotic motion, unbounded rotational motion, impact modulation motion and ship capsizing are investigated.

A novel method for response probability density of nonlinear stochastic dynamic systems

AbstractThis paper presents a novel method for analyzing high-dimensional nonlinear stochastic dynamic systems. In particular, we attempt to obtain the solution of the Fokker–Planck–Kolmogorov (FPK) equation governing the response probability density of the system without using the FPK equation directly. The method consists of several important components including the radial basis function neural networks (RBFNN), Feynman–Kac formula and the short-time Gaussian property of the response process. In the area of solving partial differential equations (PDEs) using neural networks, known as physics-informed neural network (PINN), the proposed method presents an effective alternative for obtaining solutions of PDEs without directly dealing with the equation, thus avoids evaluating the derivatives of the equation. This approach has a potential to make the neural network-based solution more efficient and accurate. Several highly challenging examples of nonlinear stochastic systems are presented in the paper to illustrate the effectiveness of the proposed method in comparison to the equation-based RBFNN approach.

Evolutionary Probabilistic Analysis of Non-Smoothly Damped Stochastic Oscillator Under Modulated Random and Harmonic Excitations

This paper concerns the evolutionary transitional probability density functions (PDFs) and the mean upcrossing rates (MCRs) of responses of the non-smoothly damped stochastic oscillator under both modulated random and harmonic excitations. The damping expressed by a non-smooth function of velocity is considered in the stochastic dynamical systems. A procedure is presented by making the time-related weighted residue of the formulated Fokker–Planck–Kolmogorov (FPK) equation vanish in the weak sense. Thus, a group of stochastic ordinary differential equations regarding the undetermined parameters in the assumed solution can be formulated and solved numerically. Three cases are studied under different conditions. The achieved evolutionary transitional PDF and MCR solutions attest to the outstanding effectiveness of the presented procedure even in the tail ends of the PDF solutions. Moreover, the achieved solutions of the stochastic dynamical systems are found to have significant non-Gaussian behaviors due to the interaction of the random and harmonic excitations and the system nonlinearity. Notably, the presented procedure greatly improves the running efficiency when compared with Monte Carlo simulation.

Threshold and Upper Bound for The Controller’s Designed Parameter of Fokker Planck Kolmogorov Probability Density Function with Applications to Cryptocurrency

This work is the first in literature to tackle the difficult open problem of determining the upper bound and threshold theorem for the TDCDP (time-dependent controller parameter) of the (Fokker Planck Kolmogorov) probability density function. This revolutionary exposition will put control theory and other related inter-disciplinary fields to a higher level towards contemporary control theory. Notably, based on the influential role of control theory in both engineering and industry, this paper will be of great value to all engineering and industry professionals who seek to know more about advanced trends within control theory settings. On the other remit of the spectrum, Fokker Planck Kolmogorov(FPK) equations are of high importance to physicists as well as mathematicians, based on their multiple applicability to information theory, graph theory, data science, finance, economics, and beyond. So, this by default adds more taste and credibility to this study. This leads by nature to introducing a different flavor to this ground-breaking research by highlighting the impact of Fokker Planck Kolmogorov(FPK) to revolutionize crypocurrency,which have received its name because it uses encryption to verify transactions, a new debatable digital payment system that doesn't rely on banks to verify transactions. It’s a peer-to-peer system that can enable anyone anywhere to send and receive payments. The paper ends with closing remarks combined with some challenging open problems and the next phase of research

Vibration Suppression of Piecewise-Linear Stiffness Nes System Under Random Excitation

Nonlinear energy sinks (NESs) are critically important for structural vibration suppression. They can absorb vibrational energy across a broad frequency spectrum, possess strong robustness, and have a relatively small mass. This study addresses the vibration suppression in a piecewise linear stiffness NES system under random excitation. Initially, a theoretical model of the piecewise linear stiffness NES system is developed. The piecewise linear stiffness function is approximated using Legendre polynomial approximation. Following this, the steady-state Fokker–Planck–Kolmogorov (FPK) equation of the system is formulated via the Generalized Harmonic Function Method. The FPK equation is solved using the fourth-order central difference method, and the effectiveness of this approach is validated by comparing the FDM results with numerical simulations. Lastly, the influence of varying system parameters on the stability of the piecewise linear stiffness NES system is analyzed.

Separable Gaussian neural networks for high-dimensional nonlinear stochastic systems

This paper extends the recently developed method of separable Gaussian neural networks (SGNN) to obtain solutions of the Fokker–Planck–Kolmogorov (FPK) equation in high-dimensional state space. Several challenges when extending SGNN to high-dimensional state space are addressed including proper definition of domain for placing Gaussian neurons and region for data sampling, and numerical integration issue of evaluating marginal probability density functions. Three benchmark nonlinear dynamic systems with increasing complexity and dimension are examined with the SGNN method. In particular, the steady-state probability density of the response is obtained with the SGNN method and compared with the results of extensive Monte Carlo simulations. It should be pointed out that some solutions of high-dimensional FPK equations for nonlinear dynamic systems would be very difficult to obtain without SGNN.

Исследование предиктивных схем управления функционированием организационно-технических систем

Organizational and technical systems are the most common tool for solving national economic problems. However, due to the personnel included in them, they are characterized by stochastic behavior, which raises the issue of developing their probabilistic model. On the other hand, the probabilistic apparatus is a tool for forecasting future phenomena and situations and therefore a mechanism for its use in predictive (forecast) management is needed. For this purpose, the work uses the first probabilistic approximation in the form of a Markov process and its description by the Fokker–Planck–Kolmogorov equation. The purpose of the study is to formulate and solve optimal control problems with probabilistic control quality criteria for a tracking predictive control scheme, as well as to develop a mechanism for metasystem switching of mo¬dels in the case of an adaptive scheme with a predictive model in the control loop. Materials and methods. Based on the analysis of scientific ideas and methodological approaches in predictive management of domestic and foreign authors, as well as mathematical methods and models, two schemes for implementing predictive management were selected to improve the efficiency of the functioning of organizational and technical systems: tracking schemes and adaptive management. Results. The optimal control problem was solved using two methods: Pontryagin's maximum principle and Bellman's principle of dynamic programming. It is shown that in the first case there is a convex upward strategy for achieving the target value of the controlled value with a maximizing approach, and in the second case the result is a convex downward strategy used for the optimal consumption of management resources. In addition, for another predictive control scheme – adaptive with a predictive model in the loop, the effectiveness of using the strategy of metasystem switching of models with the formulation and solution of six metasystem problems has been proven. Conclusion. The proposed approach can be used when designing management systems in organizational and technical systems and choosing behavior strategies in difficult market conditions when solving problems of their development.

Fully analytic approach to study bifurcation phenomenon for a shape memory alloy beam under temperature variation and lateral white-noise excitation

The main aim of this article is to obtain the probability density function of the response of an SMA beam under lateral white-noise excitation and study the bifurcation phenomenon through a fully analytic approach. Firstly, an efficient constitutive stress–strain relation for a shape memory alloy material supporting both martensite and austenite phases is considered. Secondly, the governing equation of motion for a typical SMA simply supported beam is derived using some dimensionless parameters. Thirdly, the corresponding probability density function of the response is computed analytically using the powerful Fokker–Planck–Kolmogorov equation, and finally, the bifurcation phenomenon for parameter variations of the beam is investigated. The results show how geometric (beam aspect ratio), force (the mean value of the white-noise excitation), and environmental (working temperature) factors can affect the nonlinear behavior and response bifurcation of the beam. A numerical validation is also done that guarantees the correctness of the method and results.

Non-stationary probabilistic analysis of non-linear ship roll motion due to modulated periodic and random excitations

This paper is about the study on the non-stationary probability density function (PDF) solution of the non-linear ship rolling system under modulated periodic and random excitations. The quadratic and cubic damping terms are introduced to describe the non-linear damping moment and the power terms in state variables are adopted to describe the righting moment in the ship rolling system, respectively. The relevant Fokker–Planck–Kolmogorov (FPK) equation of the ship rolling system is constructed and the non-stationary probabilistic solutions are obtained by the presented solution procedure. In the solution process, the time variable t is introduced to the traditional exponential-polynomial-closure (EPC) method to obtain the non-stationary PDF solutions of the ship rolling system. The obtained non-stationary PDF solutions of the non-linear ship rolling systems demonstrate that the presented solution procedure can give high solution accuracy even in the PDF tails when they are compared with Monte Carlo simulation (MCS). Moreover, owing to the strong system nonlinearity and the interaction of the periodic excitation and random excitation, the achieved non-stationary PDF solutions of the system responses can deviate significantly from the Gaussian PDF. It is also found that the PDFs of responses are not even symmetrical about the means of responses due to the interaction of the periodic excitation and the random excitation even if the system only contains odd nonlinear terms. In addition, the computational efficiency is increased by over five hundred times by the presented solution procedure compared to MCS.

A candidate method for prediction of the non-stationary response of strongly nonlinear systems under wide-band noise excitation

This paper expands the application of the RBFNN method to the prediction of non-stationary responses in strongly nonlinear systems subjected to wide-band noises (WBNs) excitation. Specifically, the SAM is initially employed to deduce the averaged Fokker–Planck–Kolmogorov (FPK) equation. The non-stationary responses of the energy of the system under WBNs are modeled as a one-dimensional Markov diffusion process. This yields a one-dimensional FPK equation that governs the NSRs of the system subject to WBNs. Subsequently, solving the corresponding FPK equation and obtaining the system’s NSRs are accomplished with the RBFNN method. Furthermore, four indicative cases exposed to WBNs are studied to verify the proposed scheme. Comparisons with simulations demonstrate the accuracy and efficiency of the developed technique.

Reliability assessment of stochastic dynamical systems using physics informed neural network based PDEM

In the recent decade, the reliability analysis of a stochastic system coupled with the uncertainty related to the system’s parameter has attracted much attention. Probability density evolution method (PDEM) is one of the viable options that estimates the probability density function of the structural response by solving generalized density evolution equations (GDEEs). The advantage of PDEM is that it is derived based on the principle of probability conservation, where GDEEs are decoupled from the physical system. In general, GDEEs in PDEM are solved using a finite difference scheme in which the accuracy of the numerical solution depends on the number of temporal and spatial discretizations, leading to computationally inefficient for high-fidelity models. With this in view, this study proposes a physics-informed neural network (PINN), a novel deep learning method, based PDEM, for solving the GDEEs. PINN utilizes physical information in the form of differential equations to enhance the performance of the neural networks. This method does not need any interpolation or coordinate transformation, which is often seen in any numerical scheme, thus the computational budget is reduced. Three numerical examples are presented in this study to illustrate the proposed PINN-based PDEM, including a Van-der-Pol oscillator subjected to Gaussian white noise, a one-storey moment resisting frame coupled with a nonlinear energy sink with negative stiffness and sliding friction, and a high-rise timber building coupled with shape memory alloy-based outriggers. The first example is utilized to show the accuracy of the proposed method by comparing results with the Fokker–Planck–Kolmogorov equation and Monte Carlo simulation. The rest two examples are investigated for estimating time-dependent probability of failure. Numerical results show that the proposed PINN-based PDEM can estimate the probability of failure efficiently.

Optimization-Oriented EPC Approach for Analyzing the Stochastic Nonlinear Oscillators with Displacement-Multiplicative and Additive Excitations

This paper presents a new approach named optimization-oriented exponential–polynomial-closure (OEPC) to study the behavior of stochastic nonlinear oscillators under both displacement-multiplicative and additive excitations that are Gaussian white noise. In addition to the original projection exponential–polynomial-closure (PEPC) solution procedure, the OEPC method provides an alternative procedure for solving the Fokker–Planck–Kolmogorov (FPK) equation. The OEPC method is formulated by constructing an objective function with the residue of FPK equation. By minimizing the objective function with a gradient-based method, the parameters in the exponential polynomial can be determined and the approximate PDF solution of the nonlinear random oscillator can be obtained. The solutions obtained through the OEPC approach are highly consistent with the available true solutions in special case or the Monte Carlo simulations (MCS). They are much more accurate than those obtained using the Gaussian closure method when the nonlinearity is strong. In addition, the OEPC method is computationally more efficient than MCS. The difference of the results from the OEPC and PEPC is compared and the advantage of the OEPC over PEPC is also shown when the system nonlinearity is strong and complicated.

A deep learning method based on prior knowledge with dual training for solving FPK equation

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.

On one problem of search for mobile target

At present the problems of control in conditions of conflict and uncertainty are especially relevant. This work deals with the problem of search for a mobile target in condition when the pursuer knows only probability distribution of the target initial state. In this paper we first address the auxiliary problem of controlled object convergence given terminal set. Herewith, its motion is described by a system of nonlinear differential equations and probability distribution of its initial state. We deduce formula for the probability of bringing the trajectory of controlled object to the terminal set at fixed moment of time. In so doing, the Fokker–Planck–Kolmogorov equation is used. In the case of the linear dynamics of the controlled object this formula takes an explicit form. In the paper, we apply this formula to study the problem of search in the case of linear dynamics of the pursuing controlled object (pursuer) and the mobile target. The pursuer starts moving from a given point and strives to get close to a given distance from the target. At the time it occurs the search is considered completed. Therewith, information on current state of the target is not available to the pursuer; however he is aware of probability distribution of its initial state. We derive sufficient conditions, under which at a certain time the pursuer can achieve its goal with certain probability and deduce formula for this probability. To this end, the idea of Pontryagin’s first direct method, based on the condition of the same name, is employed. In so doing, we use Minkowski operation of geometric subtraction, the properties of the set-valued mappings, and the measurable choice theorem. It is shown that, in the case of simple motion of the target and the Gaussian probability distribution of its initial state this probability takes its maximal value at the above mentioned time. On the sake of geometric descriptiveness this is illustra­ted with the example of simple motions on the plain.