Long-time analysis of stochastic heavy ball dynamics for convex optimization and monotone equations

In the framework of real Hilbert spaces, we study continuous in time dynamics as well as numerical algorithms for the problem of approaching the set of zeros of a single-valued monotone and continuous operator V. The starting point of our investigations is a second-order dynamical system that combines a vanishing damping term with the time derivative of V along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. Our method exhibits fast convergence rates of order o1tβ(t) for ‖V(z(t))‖, where z(·) denotes the generated trajectory and β(·) is a positive nondecreasing function satisfying a growth condition, and also for the restricted gap function, which is a measure of optimality for variational inequalities. We also prove the weak convergence of the trajectory to a zero of V. Temporal discretizations of the dynamical system generate implicit and explicit numerical algorithms, which can be both seen as accelerated versions of the Optimistic Gradient Descent Ascent (OGDA) method for monotone operators, for which we prove that the generated sequence of iterates (zk)k≥0 shares the asymptotic features of the continuous dynamics. In particular we show for the implicit numerical algorithm convergence rates of order o1kβk for ‖V(zk)‖ and the restricted gap function, where (βk)k≥0 is a positive nondecreasing sequence satisfying a growth condition. For the explicit numerical algorithm, we show by additionally assuming that the operator V is Lipschitz continuous convergence rates of order o1k for ‖V(zk)‖ and the restricted gap function. All convergence rate statements are last iterate convergence results; in addition to these, we prove for both algorithms the convergence of the iterates to a zero of V. To our knowledge, our study exhibits the best-known convergence rate results for monotone equations. Numerical experiments indicate the overwhelming superiority of our explicit numerical algorithm over other methods designed to solve monotone equations governed by monotone and Lipschitz continuous operators.

In the present article, the similarity method is formulated to stochastic dynamical systems described by stochastic differential equations of Stratonovich type. When a stochastic dynamical system admits symmetries, it follows that the order of stochastic equations describing the system can be reduced. Here a symmetry means an one-parameter continuous transformation which leaves the stochastic system invariant. Several examples are given, in which a nonlinear stochastic system, stochastic Hamiltonian dynamical systems related to the harmonic oscillator, and a stochastic version of the neo-classical optimal growth model suggested by Samuelson are contained.

Recent advances in data science are opening up new research fields and broadening the range of applications of stochastic dynamical systems. Considering the complexities in real-world systems (e.g., noisy data sets and high dimensionality) and challenges in mathematical foundation of machine learning, this review presents two perspectives in the interaction between stochastic dynamical systems and data science. On the one hand, deep learning helps to improve first principle-based methods for stochastic dynamical systems. AI for science, combining machine learning methods with available scientific understanding, is becoming a valuable approach to study stochastic dynamical systems with the help of observation data. On the other hand, a challenge is the theoretical explanations for deep learning. It is crucial to build explainable deep learning structures with the help of stochastic dynamical systems theory in order to demonstrate how and why deep learning works. In this review, we seek better understanding of the mathematical foundation of the state-of-the-art techniques in data science, with the help of stochastic dynamical systems, and we further apply machine learning tools for studying stochastic dynamical systems. This is achieved through stochastic analysis, algorithm development, and computational implementation. Topics involved with this review include Stochastic Analysis, Dynamical Systems, Inverse Problems, Data Assimilation, Numerical Analysis, Optimization, Nonparametric Statistics, Uncertainty Quantification, Deep Learning, and Deep Reinforcement Learning. Moreover, we emphasize available analytical tools for non-Gaussian fluctuations in scientific and engineering modeling.

In order to accelerate computations and improve long time accuracy of numerical simulations, this thesis develops multiscale geometric integrators. For general multiscale stiff ODEs, SDEs, and PDEs, FLow AVeraging integratORs (FLAVORs) have been proposed for the coarse time-stepping without any identification of the slow or the fast variables. In the special case of deterministic and stochastic mechanical systems, symplectic, multisymplectic, and quasi-symplectic multiscale integrators are easily obtained using this strategy. For highly oscillatory mechanical systems (with quasi-quadratic stiff potentials and possibly high-dimensional), a specialized symplectic method has been devised to provide improved efficiency and accuracy. This method is based on the introduction of two highly nontrivial matrix exponentiation algorithms, which are generic, efficient, and symplectic (if the exact exponential is symplectic). For multiscale systems with Dirac-distributed fast processes, a family of symplectic, linearly-implicit and stable integrators has been designed for coarse step simulations. An application is the fast and accurate integration of constrained dynamics. In addition, if one cares about statistical properties of an ensemble of trajectories, but not the numerical accuracy of a single trajectory, we suggest tuning friction and annealing temperature in a Langevin process to accelerate its convergence. Other works include variational integration of circuits, efficient simulation of a nonlinear wave, and finding optimal transition pathways in stochastic dynamical systems (with a demonstration of mass effects in molecular dynamics).

Finite dimensional Markov process approximation for stochastic time-delayed dynamical systems

The recursive estimation of states or parameters of stochastic dynamical systems with partial and imperfect measurements is generally referred to as filtering. The estimator itself is called the filter. In this dissertation optimal filters are derived for three important classes of nonlinear stochastic dynamical systems. The first class of systems, considered in Chapter II, is that governed by stochastic nonlinear hyperbolic and parabolic partial differential equations in which the dynamical disturbances in the system and in the boundary conditions can be both additive and nonadditive. This class of systems is important for it encompasses a large group of systems of practical interest, such as chemical reactors and heat exchangers. The optimal filter obtained can estimate, not only the state, but also constant parameters appearing at the boundary and in the volume of the system. The computational application of this filter is illustrated in an example of the feedback control of a styrene polymerization reactor. Many physical systems contain time delays in one form or another. Often, this kind of delay system is accompanied by some other processes such as dissipation of mass and energy, fluid mixing, and chemical reaction. In Chapter III within a single framework new optimal filters are obtained for the following classes of stochastic systems: 1. Nonlinear lumped parameter systems containing multiple constant and time-varying delays; 2. Mixed nonlinear lumped and hyperbolic distributed parameter systems; and 3. Nonlinear lumped parameter systems with functional time delays. The performance of the filter is illustrated through estimates of the temperatures in a system consisting of a well-stirred chemical reactor and an external heat exchanger. In Chapter IV filtering equations are derived for a completely general class of stochastic systems governed by coupled nonlinear ordinary and partial differential equations of either first order hyperbolic or parabolic type with both volume and boundary random disturbances. Thus, the results of Chapter III can be shown to be a special case of those obtained in Chapter IV. A related important concept to filtering is observability. For deterministic linear lumped parameter systems, observability refers to the ability to recover some prior state of a dynamical system based on partial observations of the state over some period of time. Under certain conditions, observability of the corresponding deterministic system is a sufficient condition for convergence of the optimal linear filter for a linear system with white noise disturbances. In Chapter V the concept of observability and filter convergence is developed for a class of stochastic linear distributed parameter systems whose solutions can be expressed as eigenfunction expansions. Two important questions examined are: (1) the effect of measurement locations on observability, and (2) the optimal location of measurements for state estimation.

Stochastic dynamic analysis of structures aims to explore the propagation of uncertainty in dynamic structures, referring to stochastic response and dynamic reliability analyses. For large-scale nonlinear structures, stochastic dynamic analysis is a challenging issue. In this study, a novel direct probability integral method (DPIM) is proposed to synchronously attack the problem of structural stochastic response and dynamic reliability analyses in an efficient and accurate way. The theoretical foundation of DPIM is the probability density integral equation (PDIE), an integral description of probability conservation, which decouples the evolution of probability density from the physical evolution of structure. Firstly, the PDIEs of static and dynamic structures are uniformly derived based on the principle of probability conservation, and then the equivalent differential equations are also highlighted. Then, the formula with Heaviside function for structural reliability estimation is advanced. Moreover, numerical procedures for structural stochastic responses, dynamic reliability and system reliability analyses based on DPIM are demonstrated. Finally, an example of 15-story hysteretic frame building illustrates the high efficiency and accuracy of DPIM for stochastic dynamic analysis.

The theoretical and analytical approaches of stochastic dynamics constitute the foundation for performing in‐depth inquiries into the random dynamic behavior of high‐speed railway vehicle–track system (RVTS). The proposal of the generalized probability density evolution method (PDEM) has opened up new perspectives for the stochastic dynamic analysis of complex multidimensional nonlinear systems, which is conducive to promoting the formation of a complete theoretical framework of stochastic dynamic analysis and a key technology system for the RVTS. However, as a high‐dimensional, nonlinear, and strongly time‐varying stochastic dynamics system, if a unified probability density calculation interval and a constant spatial‐integration step are used in solving the generalized density evolution equation (GDEE) of the high‐speed railway vehicle–track stochastic system, it will inevitably lead to a waste of computational memory and efficiency. Therefore, based on the theory of PDEM, this paper establishes a stochastic dynamic analysis model for the RVTS. Most importantly, an adaptive optimization of the probability density calculation interval and spatial‐integration step in the solution process is conducted, and an adaptive PDEM is developed, which provides a precise and efficient research approach for the stochastic dynamic behavior evolution analysis of the RVTS, achieving a profound balance between computational accuracy and computational efficiency. Moreover, while ensuring the accuracy, the calculation efficiency has been increased by more than 20%. Subsequently, the parametric sensitivity analysis, probability density evolution analysis, and the impact analysis of vehicle speed on the stochastic dynamic performance of the RVTS are carried out with the adaptive‐PDEM. The results show that the random vehicle loads are the most sensitive parameters in the vehicle system, which are greater than those of the stiffness and damping of the primary and secondary suspension. The vehicle speed is another important factor, and when the vehicle speed increases from 350 to 480 km/h, the vertical accelerations of the wheelsets and rails increase to ~2 times and 3 times, respectively. Additionally, some significant conclusions are obtained.

Available methods for identification of stochastic dynamical systems from input–output data generally impose restricting structural assumptions on either the noise structure in the data-generating system or the possible state probability distributions. In this paper, we introduce a novel identification method of such systems, which results in a dynamical model that is able to produce the time-varying output distribution accurately without taking restrictive assumptions on the data-generating process. The method is formulated by first deriving a novel and exact representation of a wide class of nonlinear stochastic systems in a so-called meta-state–space form, where the meta-state can be interpreted as a parameter vector of a state probability function space parameterization. As the resulting representation of the meta-state dynamics is deterministic, we can capture the stochastic system based on a deterministic model, which is highly attractive for identification. The meta-state–space representation often involves unknown and heavily nonlinear functions, hence, we propose an artificial neural network (ANN)-based identification method capable of efficiently learning nonlinear meta-state–space models. We demonstrate that the proposed identification method can obtain models with a log-likelihood close to the theoretical limit even for highly nonlinear, highly stochastic systems.

The present article focuses on the three topics related to the notions of "conserved quantities" and "symmetries" in stochastic dynamical systems described by stochastic differential equations of Stratonovich type. The first topic is concerned with the relation between conserved quantities and symmetries in stochastic Hamilton dynamical systems, which is established in a way analogous to that in the deterministic Hamilton dynamical theory. In contrast with this, the second topic is devoted to investigate the procedures to derive conserved quantities from symmetries of stochastic dynamical systems without using either the Lagrangian or Hamiltonian structure. The results in these topics indicate that the notion of symmetries is useful for finding conserved quantities in various stochastic dynamical systems. As a further important application of symmetries, the third topic treats the similarity method to stochastic dynamical systems. That is, it is shown that the order of a stochastic system can be reduced, if the system admits symmetries. In each topic, some illustrative examples for stochastic dynamical systems and their conserved quantities and symmetries are given.

The dynamics of transitions between the cells of a finite phase-space partition is analyzed for deterministic and stochastic dynamical systems in continuous time. Special emphasis is placed on the dependence of mean recurrence time on the resolution \ensuremath{\tau} between successive observations, in the limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\tau}}0.$ In deterministic systems the limit is found to exist, and to depend on only the intrinsic parameters of the underlying dynamics. In stochastic systems two different cases are identified, leading to a \ensuremath{\tau}-independent behavior and a ${\ensuremath{\tau}}^{1/2}$ behavior, depending on whether a finite speed of propagation of the signals exists or not. An extension of the results to the second moment of the recurrence time is outlined.

Stochastic dynamical systems, influenced by random disturbances, exhibit complex behaviors that are critical to understanding in various fields, such as physics, biology, and finance. The study of escape phenomena, where systems transition from stable states under the influence of noise, is essential for analyzing the dynamical behavior and learning the stochastic dynamics. This paper focuses on two key deterministic quantities, the mean exit time and the escape probability, which are widely used to analyze escape characteristics in stochastic dynamical systems. Traditional methods for computing escape problems, such as the finite difference method, finite element method, finite volume method, and Monte Carlo simulations, face challenges in high-dimensional systems and irregular domains. To address these limitations, we propose a comprehensive framework based on physics-informed neural networks. This framework is designed to solve both forward and inverse problems of escape phenomena in stochastic systems driven by Brownian motion. Our approach eliminates the need for mesh generation and naturally accommodates irregular domains, achieving a better balance between computational efficiency and accuracy. It not only overcomes the drawbacks of traditional numerical methods in solving mean exit time and escape probability but also enables learning stochastic dynamics from escape data. Through a series of numerical examples, we demonstrate the effectiveness and accuracy of the proposed method.

Semilinear stochastic dynamic systems in a separable Hilbert space often model some evolution phenomena arising in physics and engineering. In this paper, we study the existence and uniqueness of mild solutions to neutral semilinear stochastic functional dynamic systems under local non-Lipschitz conditions on the coefficients by means of the stopping time technique. We especially generalize and improve the results that appeared in Govinadan (2005), Bao and Hou (2010), and Jiang and Shen (2011).

Normal form transforms separate slow and fast modes in stochastic dynamical systems

A novel method for the dynamic reliability analysis of slopes considering dependent random parameters via the direct probability integral method