Stochastic Dynamical Systems Research Articles

Integration of multiple single cell microglial datasets reveals radial differentiation into long‐lived substates associated with Alzheimer’s pathology

AbstractBackgroundUnsupervised classification of brain cell types can be readily achieved through analysis of gene expression patterns in multidimensional space. However, the small transcriptional differences between microglia subtypes require more principled, robust, and reproducible approaches to expose and distinguish biological differences from noise.MethodHere we leverage Fokker‐Planck diffusion maps [1,2] to examine the dynamics of microglia subtypes by finding a low dimensional representation of the underlying stochastic dynamical system responsible for generating these transcriptional states. Clustering directly on this projection allows us to reduce noise, assign time scales to these substates, and generate a hierarchy of microglial cell states. Once these subtypes have been defined, we apply pseudobulk differential expression analysis to identify marker genes, and use those genes to build a boosted tree model to classify cells states. We then apply Shapley additive explanations (SHAP) to identify which genes are most important for distinguishing between cell states.ResultWe observe a radial transition pattern emanating, with a subtype that appears homeostatic, inactivated, and poised for transition. This is consistent with the observation that microglia dynamics are sensitive to their microenvironment. Deconvolution of an independent bulk microglia dataset (n = 130) showed depletion in subtypes linked to protein misfolding and myelin phagocytosis, as well as enrichment in immunoreactive microglia in Alzheimer’s patients. SHAP reveals DUSP1, CD83, PLCG2, OLR1,CCL2, and JUN among the most important genes for differentiating into these AD‐associated clusters.ConclusionOur approach enables us to reproducibly identify microglial subtypes across sets of single cell‐resolution high dimensional molecular profile data, identifying likely drivers of subtype differentiation. With several of our driver genes already associated with Alzheimer’s via GWAS and other analyses, our results will enable us to elucidate the role of these genes in determining microglial subtypes.

Filtering in Triplet Markov Chain Model in the Presence of Non-Gaussian Noise with Application to Target Tracking

In hidden Markov chain (HMC) models, widely used for target tracking, the process noise and measurement noise are in general assumed to be independent and Gaussian for mathematical simplicity. However, the independence and Gaussian assumptions do not always hold in practice. For instance, in a typical radar tracking application, the measurement noise is correlated over time as the sampling frequency of a radar is generally much higher than the bandwidth of the measurement noise. In addition, target maneuvers and measurement outliers imply that the process noise and measurement noise are non-Gaussian. To solve this problem, we resort to triplet Markov chain (TMC) models to describe stochastic systems with correlated noise and derive a new filter under the maximum correntropy criterion to deal with non-Gaussian noise. By stacking the state vector, measurement vector, and auxiliary vector into a triplet state vector, the TMC model can capture the complete dynamics of stochastic systems, which may be subjected to potential parameter uncertainty, non-stationarity, or error sources. Correntropy is used to measure the similarity of two random variables; unlike the commonly used minimum mean square error criterion, which uses only second-order statistics, correntropy uses second-order and higher-order information, and is more suitable for systems in the presence of non-Gaussian noise, particularly some heavy-tailed noise disturbances. Furthermore, to reduce the influence of round-off errors, a square-root implementation of the new filter is provided using QR decomposition. Instead of the full covariance matrices, corresponding Cholesky factors are recursively calculated in the square-root filtering algorithm. This is more numerically stable for ill-conditioned problems compared to the conventional filter. Finally, the effectiveness of the proposed algorithms is illustrated via three numerical examples.

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.

The most likely transition path for a class of distribution-dependent stochastic systems

Distribution-dependent stochastic dynamical systems arise widely in engineering and science. We consider a class of such systems which model the limit behaviors of interacting particles moving in a vector field with random fluctuations. We aim to examine the most likely transition path between equilibrium stable states of the vector field. In the small noise regime, the action functional does not involve the solution of the skeleton equation which describes the unperturbed deterministic flow of the vector field shifted by the interaction at zero distance. As a result, we are led to study the most likely transition path for a stochastic differential equation without distribution dependency. This enables the computation of the most likely transition path for these distribution-dependent stochastic dynamical systems by the adaptive minimum action method and we illustrate our approach in two examples.

A new result on the fractal dimension estimates of random attractor for non-autonomous random 2D stochastic dynamical type systems

We prove some conditions for bounding the fractal dimension of random invariant sets of non-autonomous random dynamical systems on separable Banach spaces. Then we apply these conditions to prove the finiteness of fractal dimension of random attractor for stochastic 2D hydrodynamical type equations with linear additive white noise in bounded domains or unbounded domains satisfying the Poincaré inequality.

Global sensitivity evolution equation of the Fréchet-derivative-based global sensitivity analysis

For stochastic dynamical systems with multiple uncertain parameters, it is often of interest to detect which parameters are dominant, in which the global sensitivity analysis may be one of the common means. To measure the global sensitivity in both qualitative and quantitative terms, it is of significant importance to adopt a global sensitivity index with sufficient quantification information. The Fréchet-derivative-based global sensitivity index (Fre-GSI) proposed by Chen et al. (2020) is appropriate to this goal. The present paper aims to provide new aspects of the Fre-GSI, including: (1) The numerical solution of the Fre-GSI given by Chen et al. (2020) is investigated in both analytical and numerical aspects; (2) A novel global sensitivity evolution equation is derived from the generalized density evolution equation, thus the Fre-GSI can be estimated by directly solving the global sensitivity evolution equation, rather than repeatedly solving the generalized density evolution equation as suggested in Chen et al. (2020). Numerical examples are studied to illustrate the efficiency and accuracy of the proposed approach. Some problems to be further studied are also outlined.

Deep reinforcement learning in finite-horizon to explore the most probable transition pathway

In many scientific and engineering problems, noise and nonlinearity are unavoidable, which could induce interesting mathematical problem such as transition phenomena. This paper focuses on efficiently discovering the most probable transition pathway for stochastic dynamical systems employing reinforcement learning. With the Onsager–Machlup action functional theory to quantify rare events in stochastic dynamical systems, finding the most probable pathway is equivalent to solving a variational problem on the action functional. When the action function cannot be explicitly expressed by paths near the reference orbit, the variational problem needs to be converted into an optimal control problem. First, by integrating terminal prediction into the reinforcement learning framework, we develop a Terminal Prediction Deep Deterministic Policy Gradient (TP-DDPG) algorithm to deal with the finite-horizon optimal control issue in a forward way. Next, we present the convergence analysis of our algorithm for the value function in terms of the neural network’s approximation error and estimation error. Finally, we conduct various experiments in different dimensions for the transition problems in applications to illustrate the effectiveness of our algorithm.

Weak averaging principle for multiscale stochastic dynamical systems driven by stable processes

We study the averaging principle for a family of multiscale stochastic dynamical systems. The fast and slow components of the systems are driven by two independent stable Lévy noises, whose stable indexes may be different. The homogenizing index r0 of slow components has a relation with the stable index α1 of the noise of fast components given by 0<r0<2−2/α1. By first studying a nonlocal Poisson equation and then constructing suitable correctors, we obtain that the slow components weakly converge to a Lévy process as the scale parameter goes to zero.

Multiple adaptive fuzzy Nussbaum-type functions design for stochastic nonlinear systems with fixed-time performance

Nussbaum design has been successfully exploited to deal with uncertain systems with a single unknown control coefficient. However, it is challenging for the traditional Nussbaum method to efficiently handle the feedback control design with multiple unknown time-varying control coefficients because the multiple Nussbaum-type gains might mutually cancel out, which makes the Lyapunov-based stability analysis infeasible. To tackle this issue, a novel type of Nussbaum function is designed for the tracking problem of a class of stochastic strict-feedback nonlinear systems. In the presented Nussbaum design, the multiple unknown control coefficients are separately compensated, where the effect of mutual cancellation no longer exists and the Lyapunov stability analysis can be directly applied. Moreover, the proposed Nussbaum design is combined with the fixed-time stability theory, which ensures the settling time is bounded and independent of the initial condition. Within the adaptive fuzzy backstepping framework, it is shown that all signals in the closed-loop system remain bounded through the theoretical discussions. The simulation experiment is conducted to verify the effectiveness of the presented design for the continuous-time stochastic nonlinear dynamical system.

Application of Migrating Optimization Algorithms in Problems of Optimal Control of Discrete-Time Stochastic Dynamical Systems

The problem of finding the optimal open-loop control for discrete-time stochastic dynamical systems is considered. It is assumed that the initial conditions and external influences are random. The average value of the Bolza functional defined on individual trajectories is minimized. It is proposed to solve the problem by means of classical and modified migrating optimization algorithms. The modification of the migrating algorithm consists of cloning the members of the initial population and choosing different strategies of migratory behavior for the main population and for populations formed by clones. At the final stage of the search for an extremum, an intensively clarifying migration cycle is implemented with the participation of three leaders of the populations participating in the search process. Problems of optimal control of bundles of trajectories of deterministic discrete dynamical systems, as well as individual trajectories, are considered as special cases. Seven model examples illustrating the performance of the proposed approach are solved.

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.

DeepBayes—An estimator for parameter estimation in stochastic nonlinear dynamical models

Stochastic nonlinear dynamical systems are ubiquitous in modern, real-world applications. Yet, estimating the unknown parameters of stochastic, nonlinear dynamical models remains a challenging problem. The majority of existing methods employ maximum likelihood or Bayesian estimation. However, these methods suffer from some limitations, most notably the substantial computational time for inference coupled with limited flexibility in application. In this work, we propose DeepBayes estimators that leverage the power of deep recurrent neural networks. The method consists of first training a recurrent neural network to minimize the mean-squared estimation error over a set of synthetically generated data using models drawn from the model set of interest. The a priori trained estimator can then be used directly for inference by evaluating the network with the estimation data. The deep recurrent neural network architectures can be trained offline and ensure significant time savings during inference. We experiment with two popular recurrent neural networks — long short term memory network (LSTM) and gated recurrent unit (GRU). We demonstrate the applicability of our proposed method on different example models and perform detailed comparisons with state-of-the-art approaches. We also provide a study on a real-world nonlinear benchmark problem. The experimental evaluations show that the proposed approach is asymptotically as good as the Bayes estimator.

A decoupled approach for determination of the joint probability density function of a high-dimensional nonlinear stochastic dynamical system via the probability density evolution method

The joint probability density function (PDF) of multiple response processes of a system is a crucial topic in the fields of science and engineering. It is adopted to describe the dependent uncertainty propagation in complex stochastic dynamical systems, including the displacement and velocity of one degree of freedom (DOF), and the nonlinear probabilistic dependencies between multiple variables. However, the partial differential equations (PDEs) governing the time evolution of joint PDFs for multiple stochastic processes are often high-dimensional and coupled, making them analytically or numerically intractable for systems with high dimensionality. The probability density evolution method (PDEM) has provided a new perspective from the standpoint of random events description of the principle of preservation of probability. For an n-dimensional system, if merely the joint PDF of m processes is needed, where m is not larger than n, then it is sufficient to solve the generalized density evolution equation (GDEE), i.e., the Li–Chen equation, in m dimensions. This approach is particularly convenient for cases where n is large while m is small. However, for cases where m is not less than two, it is still challenging to avoid solving high-dimensional PDEs. This paper proposes a new method for the determination of joint PDF of multiple responses based on the principle of preservation of probability. To capture the joint PDF of m processes, the m-dimensional PDE can be transformed into m one-dimensional PDEs, significantly reducing the computational complexity of numerical implementation. The proposed method gives an enhanced version to the PDEM, enabling it to not only determine the individual probabilistic response, but also efficiently determine the dependent probabilistic distribution of multiple responses of high-dimensional stochastic dynamical systems without solving the coupling PDE. It reduces the computational cost from exponential growth of m powers to linear growth. Some numerical examples are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.

Sharp asymptotic estimates for expectations, probabilities, and mean first passage times in stochastic systems with small noise

AbstractFreidlin‐Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well‐posed matrix Riccati equations involving the minimizer of the Freidlin‐Wentzell action as input, either forward or backward in time with appropriate initial or final conditions tailored to the estimate at hand. The usefulness of our approach is illustrated on several examples. In particular, invariant measure probabilities and mean first passage times are calculated in models involving stochastic partial differential equations of reaction‐advection‐diffusion type.

General validity of the second fluctuation-dissipation theorem in the nonequilibrium steady state: Theory and applications

In this paper, we derive a generalized second fluctuation-dissipation theorem (FDT) for stochastic dynamical systems in the steady state and further show that if the system is highly degenerate, then the classical second FDT is valid even when the exact form of the steady state distribution is unknown. The established theory is built upon the Mori-type generalized Langevin equation for stochastic dynamical systems and hence generally applies to nonequilibrium systems driven by stochastic forces. These theoretical results enable us to construct a data-driven nanoscale fluctuating heat conduction model based on the second FDT. We numerically verify that our heat transfer model yields better predictions than the Green-Kubo formula for systems far from the equilibrium.

Grid-Free Computation of Probabilistic Safety With Malliavin Calculus

This work concerns continuous-time, continuous-space stochastic dynamical systems described by stochastic differential equations (SDE). It presents a new approach to compute probabilistic safety regions, namely sets of initial conditions of the SDE associated to trajectories that are safe with a probability larger than a given threshold. The approach introduces a functional that is minimised at the border of the probabilistic safety region, then solves an optimisation problem using techniques from Malliavin Calculus, which computes such region. Unlike existing results in the literature, the new approach allows one to compute probabilistic safety regions without gridding the state space of the SDE.

Finite time SF bifurcation and stability analysis for stochastic time-varying delay system

In this paper, it is found that the systems alternate between oscillatory and steady states in studying the stability of a class of stochastic dynamical systems with fast time-varying periodic delays. We propose a new definition of bifurcation, State Flip (SF) bifurcation, by using the abrupt change in the dwell time ratio of two states. Taking the amplitude of the time-varying delay function δ as the bifurcation parameter, we investigate the dynamics of the evolution of a time-varying delay system driven by Gaussian white noise. Firstly, we convert systems with fast time-varying periodic delays into comparative systems with distributed delays, which in turn are converted into stochastic systems with constant delays for discussion. In order to reduce the stochastic delay equation to the average Itô equation, we adopted the stochastic averaging method and the generalized central manifold theory. It is used to capture the one-dimensional slow variables of a system near the critical state of the transition between two steady states during the evolution of the system. We then find that as the amplitude increases from weak to strong, the stochastic time-varying delay system changes from a two-period oscillation to alternating between oscillation and steady, and finally to a steady state. Lastly, we perform numerical simulations of the original stochastic time-varying delay system and the transformed two-dimensional stochastic constant time-delay system respectively to verify that the ideas and methods in this paper are reasonable and effective.

Reservoir computing with error correction: Long-term behaviors of stochastic dynamical systems

The prediction of stochastic dynamical systems and the capture of dynamical behaviors are profound problems. In this article, we propose a data-driven framework combining Reservoir Computing and Normalizing Flow to study this issue, which mimics error modeling to improve traditional Reservoir Computing performance and integrates the virtues of both approaches. With few assumptions about the underlying stochastic dynamical systems, this model-free method successfully predicts the long-term evolution of stochastic dynamical systems and replicates dynamical behaviors. We verify the effectiveness of the proposed framework in several experiments, including the stochastic Van der Pal oscillator, El Niño-Southern Oscillation simplified model, and stochastic Lorenz system. These experiments consist of Markov/non-Markov and stationary/non-stationary stochastic processes, which are defined by linear/nonlinear stochastic differential equations or stochastic delay differential equations. Additionally, we explore the noise-induced tipping phenomenon, relaxation oscillation, stochastic mixed-mode oscillation, and replication of the strange attractor.

A new path integration method for the stochastic system under Poisson white noise excitation based on a probability mapping

In this paper, a new path integration method is proposed for stochastic dynamical systems excited by Poisson white noise. An efficient one-step transition probability density function (TPDF) matrix is constructed based on a decoupling probability mapping. The new method can handle the case of multiple impulses occurring in a one-step transition time interval and considers the randomness of the impulse instant, which compensates for a drawback of previous path integration methods. The probability mapping realizes the decoupling of randomness and the one-step TPDF matrix, which can be extended to general stochastic systems satisfying the Markov property. The stochastic responses of two dynamical systems excited by Poisson white noises for different mean arrival rates are obtained by using the new path integration method, and MC simulations prove that the new method is very effective, and it maintains good accuracy even for large mean arrival rates.

Weak Pullback Mean Attractor for \(p\)-Laplacian Selkov Lattice Systems with Locally Lipschitz Delay Diffusion Terms

This paper focuses on the dynamics of a class of nonlinear, reversible, random \(p\)-Laplace Selkov delay lattice systems defined by local lipschitz noise-driven \(\mathbb{Z}^d\). We first establish the global fitness of the system using the local Lipschitz delayed diffusion term. Under certain conditions, we demonstrate the existence and uniqueness of the mean stochastic dynamical system in relation to the stochastic equation in the product Hilbert space \(L^2(\Omega, \mathcal{F}_\tau; \ell^2\times\ell^2） \times L^2 （\Omega， \mathcal{F}_\tau; L^2((-\rho, 0), \ell^2\times\ell^2) \). The average stochastic dynamical system theory proposed by Wang (J.Equ., 31:2177-2204, 2019) is used to deal with the difficulties caused by nonlinear noise. Even if the discrete \(p\)-Laplace is replaced by the usual discrete Laplace, the results of this paper are new.