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Van der Pol system in the stationary state under random additive excitation (multiplicative excitation prevented); lower linear damping -part (a), higher linear damping -part (b).
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The problems that often arise in stochastic dynamics can be investigated using the Fokker–Planck (FP) equation. The response of a such systems being subjected to additive and/or multiplicative random noise is represented by probability density function (PDF) that gives the full information about a response random character. Various analytic and sem...
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... Even for systems with exact solutions, there are always conditions that cannot be satisfied in practice. Therefore, numerical methods such as the finite difference method [6], finite element method [7,8], generalized and compatible cell mapping method [9,10], and path integration method [11,12] have been studied extensively. Nevertheless, as the dimensionality increases, the computing and storage resources of these methods may in-crease exponentially. ...
Nonlinear random vibration is a common phenomenon, and predicting its probability density is an essential component of vibration engineering. This paper proposes a data-driven method for identifying explicit expressions of response probability densities in random vibrating systems with implicit Hamiltonian functions. The process concludes with two steps, identifying the Hamiltonian function and calculating the probability density of the stationary response. The former uses the differential equations of the motion of the quasi-Hamiltonian systems to identify Hamiltonian functions from the simulated data, while the latter estimates the logarithm of the probability density from the identified Hamiltonian functions and acquires an explicit expression. Their unknown coefficients can be attributed to the solution of a set of undetermined equations. The proposed method is applied to systems in which the Hamiltonian functions cannot be simply derived, such as those with complicated stiffness. Two examples are presented to demonstrate the applicability and effectiveness of our method, i.e., the nonlinear vibration energy harvester (VEH) and the Duffing oscillator with LuGre friction. The proposed technique outperforms Monte Carlo simulations (MCSs) in efficiency. The results show that our method is insensitive to parameters and can be used for identifying transient probability density. Its application scope is wider than the stochastic averaging method. Data-driven method, Probability density, Implicit Hamiltonian function, Vibration energy harvester, LuGre friction Citation: Y. Chen, S. Wang, and G. Jiao, Two-step data-driven identification of probability densities for random vibrating systems with implicit Hamiltonian functions, Acta Mech. Sin. 40, 523459 (2024), https://doi.
... Other approximate and numerical approaches for solving the FPK equation, including equivalent linearization [6][7][8][9], equivalent nonlinearization [10][11][12], stochastic averaging method [13,14], closure method [15], finite element method [16], finite difference method [17], path integral method [18][19][20], the exponential polynomial closure method for the stationary PDF of nonlinear systems [21], the statespace-split exponential polynomial closure method for high dimensional nonlinear systems [22], the generalized cell mapping method [23] and Monte Carlo simulation [24]. All these methods have various limitations when applied to study transient and stationary PDF of high-dimensional nonlinear stochastic systems due to the excessive demand on memory and CPU time needed to find the global solution of the PDF in the high-dimensional state space. ...
... See 9.2 for a description of the Monte Carlo algorithm. [4], [61], [53] and finite element [40], [39] methods. For a comparison of these traditional methods the reader can look at this comparative study [45] by Pitcher et al where the methods have been applied to 2 and 3 dimensional examples. ...
In this paper we devise a deep learning algorithm to find non-trivial zeros of Fokker-Planck operators when the drift is non-solenoidal. We demonstrate the efficacy of our algorithm for problem dimensions ranging from 2 to 10. Our method scales linearly with dimension in memory usage. We show that this method produces better approximations compared to Monte Carlo methods, for the same overall sample sizes, even in low dimensions. Unlike the Monte Carlo methods, our method gives a functional form of the solution. We also demonstrate that the associated loss function is strongly correlated with the distance from the true solution, thus providing a strong numerical justification for the algorithm. Moreover, this relation seems to be linear asymptotically for small values of the loss function.
... The discretization methods for solving the FP equations are strongly depended on meshing, resulting in an exponential increase in computation with dimension, for instance, finite element [9,17], finite difference [13,20], path integral [5,26]. In particular, it must make a trade-off between precision and hardware storage capacity when dealing with high-dimensional cases. ...
... When solving Eq. (18), the first step is still to obtain the training points which can be done by applying the SRKII method for Eq. (17). In order to further analyze and verify the performance of deep KD-tree, the data ratio coefficient is still not set. ...
... Therefore, it can be found from the analysis of Example 3 and Example 2 that in the actual use of deep KD-tree, it is reasonable to set a data ratio coefficient. At the same time, the results in Fig. 9 again show (a) (b) Fig. 9 a The integral contribution rate of different data sets increases with the depth of deep KD-tree for system (17). b The utilization rate of different data sets increases with the depth of deep KD-tree for system (17). ...
The Fokker–Planck (FP) equation can deterministically describe the evolution of the probability density function, which plays an extremely significant role in the fields of stochastic dynamics. Unfortunately, the limited samples that arise from the consideration of engineering practice are inevitable, which restricts the solving of the FP equation. Accordingly, in the present study, a super-DL-FP framework is established to solve the steady-state FP equation with a small amount of data, through combining the deep KD-tree and the DL-FP approach proposed in [Chaos 30, 013133 (2020)]. It should be emphasized that the normalization condition is of great importance and has to be considered in solving the steady-state FP equation. An appropriate integral estimation for the normalization condition under non-uniform meshing can effectively improve the precision of the solution, but it is still a challenging problem, especially for the case of small data. Thus, the so-called deep KD-tree method is innovatively proposed to estimate the normalized integral with a small random dataset. The main target is to obtain the appropriate discrete integral points and corresponding integral volumes by executing multiple KD-tree segmentation based on random data on the integral region. Several numerical experiments and comparisons are implemented to illustrate the superior performance of the super-DL-FP method. The obtained results indicate that the proposed algorithm can accomplish higher accuracy in the sense of lower cost than the well-known algorithms like center difference scheme, Chebyshev spectrum algorithm, and normalized flow approach.
... The probability transformation method to obtain the probability density function (PDF) was introduced by Falsone and Settineri [17][18][19]. For more complex nonlinear systems, the statistical linearization [20], equivalent nonlinear system [21], path integral method [22,23], stochastic averaging method [24], Hamiltonian system method [25], and some other numerical techniques [26,27] were developed to resolve the corresponding problem. However, these methods may not be always applicable for the strong nonlinear system [4]. ...
This paper proposes a fully adaptive method to estimate the probability density function (PDF) of the responses for stochastic structures under static and dynamic loads. The PDF is described by a recently published direct probability integral method (DPIM). In DPIM, the key smoothing parameter σ and the number of samples Ns, which have significant impacts on the determination of PDF and the efficiency of stochastic response calculation, are still difficult to determine in a completely adaptive manner. To this end, we derive a weighted kernel density estimation (weighted-KDE) method to adaptively obtain the σ by minimizing the mean integrated squared error between the estimated PDF and the real PDF. In addition, a new iterative sampling strategy is proposed to adaptively determine the Ns, in which a candidate sample pool is generated firstly, then the samples are gradually selected from the pool. This strategy ensures that the used samples in each iteration can fill the probability space in a ‘highly uniform’ manner. Based on the standard deviation calculated by the estimated PDF, a stopping criterion is presented to stop the iterative process. Moreover, the detailed steps of the proposed fully adaptive method is given by combining the weighted-KDE method and the sampling strategy. Four engineering examples validate the adaptive ability, efficiency and accuracy of the proposed method.
... The discretization methods for solving the FP equations are strongly depended on meshing, resulting in an exponential increase in computation with dimension, for instance, finite element [17,9], finite difference [13,20], path integral [5,26]. In particular, it must make a trade-off between precision and hardware storage capacity when dealing with high-dimensional cases. ...
The Fokker-Planck (FP) equation can deterministically describe the evolution of the probability density function, which plays an extremely significant role in the fields of stochastic dynamics. Unfortunately, the limited samples that arise from the consideration of engineering practice are inevitable, which restricts the solving of the FP equation. Accordingly, in the present study, a super-DL-FP framework is established to solve the steady-state FP equation with a small amount of data, through combining the deep KD-tree and the DLFP approach proposed in [Chaos 30, 013133 (2020)]. It should be emphasized that the normalization condition is of great importance and have to be considered in solving the steady-state FP equation. An appropriate integral estimation for the normalization condition under non-uniform meshing can effectively improve the precision of the solution, but it is still a challenging problem, especially for the case of small data. Thus, the so-called deep KD-tree method is innovatively proposed to estimate the normalized integral with a small random dataset. The main target is to obtain the appropriate discrete integral points and corresponding integral volumes by executing multiple KD-tree segmentation based on random data on the integral region. Several numerical experiments and comparisons are implemented to illustrate the superior performance of the super-DL-FP method. The obtained results indicate that the proposed algorithm can accomplish higher accuracy in the sense of lower cost than the well-known algorithms like center difference scheme, Chebyshev spectrum algorithm, and normalized flow approach.
... There are several numerical techniques to solve the FPKE like the finite element method [7][8][9], finite difference method [10,11], path integral methods [12,13], that have emerged from the literature on solving PDEs. While these methods perform reliably for systems with low dimensions , they do not scale well and are known to become computationally intractable for high-dimensional systems. ...
View Video Presentation: https://doi.org/10.2514/6.2022-1441.vid In this paper, we seek to estimate the invariant sets of stochastic dynamical systems by solving the steady-state form of the Fokker-Planck-Kolmogorov equation (FPKE) using Physics-Informed Neural Networks (PINNs). It is a second-order partial differential equation (PDE) that is challenging to solve numerically because of the positivity and normalization constraints imposed on the probability density function (pdf) variable. We propose an alternative formulation that transforms the PDE into an equivalent form in which the pdf conditions are either already satisfied, or can be addressed in post-processing. This allows us to use the conventional PINNs framework to solve the resulting PDE . We illustrate the validity of the solution on a few examples.
... There are several numerical techniques to solve the FPKE like the finite element method [8][9][10], finite difference method [11,12], path integral methods [13,14], that have emerged from the literature on solving PDEs. While these methods perform reliably for systems with low dimensions , they do not scale well and are known to become computationally intractable for high-dimensional systems. ...
In this paper, we propose a refinement strategy to the well-known Physics-Informed Neural Networks (PINNs) for solving partial differential equations (PDEs) based on the concept of Optimal Transport (OT). Conventional black-box PINNs solvers have been found to suffer from a host of issues: spectral bias in fully-connected architectures, unstable gradient pathologies, as well as difficulties with convergence and accuracy. Current network training strategies are agnostic to dimension sizes and rely on the availability of powerful computing resources to optimize through a large number of collocation points. This is particularly challenging when studying stochastic dynamical systems with the Fokker-Planck-Kolmogorov Equation (FPKE), a second-order PDE which is typically solved in high-dimensional state space. While we focus exclusively on the stationary form of the FPKE, positivity and normalization constraints on its solution make it all the more unfavorable to solve directly using standard PINNs approaches. To mitigate the above challenges, we present a novel training strategy for solving the FPKE using OT-based sampling to supplement the existing PINNs framework. It is an iterative approach that induces a network trained on a small dataset to add samples to its training dataset from regions where it nominally makes the most error. The new samples are found by solving a linear programming problem at every iteration. The paper is complemented by an experimental evaluation of the proposed method showing its applicability on a variety of stochastic systems with nonlinear dynamics.
... There are several available numerical schemes to investigate the FP equations. for instance the discrete scheme and the typical methods include path integration [18,19], finite element [20,21], and finite difference [22,23]. However, it is unfortunate that these methods are limited by the size of the grid and can only solve the points on the grid. ...
The probability density function of the stochastic differential equations with α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-stable Lévy noise is determined by the deterministic fractional Fokker–Planck (FFP) equation with the Riesz fractional derivative. In this paper, to solve the FFP equation, a new algorithm which is an efficient numerical approach using the deep neural network is proposed in view of the mesh-dependence phenomenon in the traditional discrete scheme and the nonsmooth solution of the Monte Carlo method. Under the framework of the DL-FP algorithm, the “fractional centered derivative” approach is applied to approximate the Riesz fractional derivative of the output in the neural network, which is the major novelty of this approach. Numerical results are presented to demonstrate the accuracy of the approach. Comparing with the traditional discrete scheme and the Monte Carlo method, the proposed algorithm is mesh-less and smoother. Furthermore, the proposed technique performs well in dealing with the heavy-tail case through comparing with the analytical solution.
... Many authors have been solved fractional Fokker-Plank equation by using iterative methods such as Yildirim [38], Kumar [39], Jawary [40], Dubet et al. [41], Santos and Gomez [42]. Few other Fokker Plank equation have been solved by using other methods such as [43][44][45][46][47][48][49][50]. The nonlinear FPE of one variable is written in the form: ...
This endeavour provides a new instance of understanding the velocity of a particle in Brownian motion, using the Fokker-Plank equation. Our treatment is based on the new mathematical tool invoked by Yang-Abdel-Aty-Cattani (YAC) to solve the fractional derivative on Fokker-Plank equation. The HPTM and RPSMs, are used to obtain the solution in terms of Prabhakar function. Numerical examples to extend these ideas and to establish the capability and truthfulness of the proposed method, which makes use of YAC-fractional derivative, is also briefly sketched.
