Non-parametric Estimation of Stochastic Differential Equations from Stationary Time-Series

Chen, Xi; Timofeyev, Ilya

doi:10.1007/s10955-021-02847-6

Cited by 5 publications

(15 citation statements)

References 23 publications

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“…where E x [•] denotes the expectation under (9), with respect to the initial condition X 0 = x. When t = δ, we have…”

Section: Assumption 23 There Is a Functionmentioning

confidence: 99%

“…In [40], Theorem 2.1 has been applied to a variety of SDEs, including the Langevin dynamics, monotone and dissipative systems, and stochastic gradient systems. The function V in Assumption 2.3 is called the Lyapunov function of the dynamical system (9). In particular, we further assume that V is of a polynomial growth rate.…”

Section: Assumption 23 There Is a Functionmentioning

confidence: 99%

“…We consider a family of Markov evolution operators {P ǫ t | t ≥ 0, ǫ ∈ (−ǫ 0 , ǫ 0 )} on R d that characterize the unperturbed dynamics in (9) and its perturbations. We will specify such P ǫ t in Section 3.…”

Section: The Long-time Linear Response Theorymentioning

confidence: 99%

“…Here, to help readers understand the notations, one can interpret the parameter ǫ as the strength of the perturbation. In other words, when ǫ = 0, P 0 t reduces to the evolution operator of the unperturbed dynamics, e.g., the Itô diffusion (9). Namely,…”

Section: The Long-time Linear Response Theorymentioning

confidence: 99%

“…Since the same problem can be posed as a supervised learning task, a lot of recent interest has been shifting to machine learning approaches. Among the linear estimators, a popular approach is the kernelbased method [9,42,17,51,35,10,11], whose connection to the parametric modeling paradigm has been studied in [31]. In this direction, many nonparametric models have been proposed, including the orthogonal polynomials [51], wavelets [42], Gaussian processes [17], radial kernels [17], diffusion maps based models [4,5,18], just to name a few.…”

Section: Introductionmentioning

confidence: 99%

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Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Itô Diffusions

Zhang,

Harlim,

2021

Preprint

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This paper studies the theoretical underpinnings of machine learning of ergodic Itô diffusions. The objective is to understand the convergence properties of the invariant statistics when the underlying system of stochastic differential equations (SDEs) is empirically estimated with a supervised regression framework. Using the perturbation theory of ergodic Markov chains and the linear response theory, we deduce a linear dependence of the errors of one-point and two-point invariant statistics on the error in the learning of the drift and diffusion coefficients. More importantly, our study shows that the usual L 2 -norm characterization of the learning generalization error is insufficient for achieving this linear dependence result. We find that sufficient conditions for such a linear dependence result are through learning algorithms that produce a uniformly Lipschitz and consistent estimator in the hypothesis space that retains certain characteristics of the drift coefficients, such as the usual linear growth condition that guarantees the existence of solutions of the underlying SDEs. We examine these conditions on two well-understood learning algorithms: the kernel-based spectral regression method and the shallow random neural networks with the ReLU activation function.

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“…where E x [•] denotes the expectation under (9), with respect to the initial condition X 0 = x. When t = δ, we have…”

Section: Assumption 23 There Is a Functionmentioning

confidence: 99%

Section: Assumption 23 There Is a Functionmentioning

confidence: 99%

Section: The Long-time Linear Response Theorymentioning

confidence: 99%

Section: The Long-time Linear Response Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Itô Diffusions

Zhang,

Harlim,

2021

Preprint

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On the anti-missile interception technique of unpowered phase based on data-driven theory

Huang

2022

Mechanics & Industry

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Abstract. The anti-missile interception technique of unpowered phase is of much importance in the military field, which depends on the prediction of the missile trajectory and the establishment of the missile model. With rapid development of data science field and large amounts of available data observed, there are more and more powerful data-driven methods proposed recently in discovering governing equations of complex systems. In this work, we introduce an anti-missile interception technique via a data-driven method based on Koopman operator theory. More specifically, we describe the dynamical model of the missile established by classical mechanics to generate the trajectorial data. Then we perform the data-driven method based on Koopman operator to identify the governing equations for the position and velocity of the missile. Numerical experiments show that the trajectories of the learned model agree well with the ones of the true model. The effectiveness and accuracy of this technique suggest that it will be realized in practical applications of anti-missile interception.

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Detecting physical laws from data of stochastic dynamical systems perturbed by non-Gaussian α-stable Lévy noise

Liu

2023

Chinese Phys. B

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Massive data from observations, experiments and simulations of dynamical models in scientific and engineering fields make it desirable for data-driven methods to extract basic laws of these models. We present a novel method to identify such high dimensional stochastic dynamical systems that perturbed by a non-Gaussian $\alpha$-stable Lévy noise. More explicitly, firstly a machine learning framework to solve the sparse regression problem is established to grasp the drift terms through one of non-local Kramers-Moyal formulas. Then the jump measure and intensity of the noise are disposed by the relationship with statistical characteristics of the process. Three examples are then given to demonstrate the feasibility. This approach proposes an effective way to understand the complex phenomena of systems under non-Gaussian fluctuations and illuminates some insights into the exploration for further typical dynamical indicators such as the maximum likelihood trasition path or mean exit time of these stochastic systems.

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Non-parametric Estimation of Stochastic Differential Equations from Stationary Time-Series

Cited by 5 publications

References 23 publications

Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Itô Diffusions

Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Itô Diffusions

On the anti-missile interception technique of unpowered phase based on data-driven theory

Detecting physical laws from data of stochastic dynamical systems perturbed by non-Gaussian α-stable Lévy noise

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