Extracting Governing Laws from Sample Path Data of Non-Gaussian Stochastic Dynamical Systems

• URL: http://arxiv.org/abs/2107.10127v1
• Date: Wed, 21 Jul 2021 14:50:36 GMT
• Title: Extracting Governing Laws from Sample Path Data of Non-Gaussian Stochastic Dynamical Systems
• Authors: Yang Li and Jinqiao Duan
• Abstract summary: We infer equations with non-Gaussian L'evy noise from available data to reasonably predict dynamical behaviors. We establish a theoretical framework and design a numerical algorithm to compute the asymmetric L'evy jump measure, drift and diffusion. This method will become an effective tool in discovering the governing laws from available data sets and in understanding the mechanisms underlying complex random phenomena.
• Score: 4.527698247742305
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Advances in data science are leading to new progresses in the analysis and understanding of complex dynamics for systems with experimental and observational data. With numerous physical phenomena exhibiting bursting, flights, hopping, and intermittent features, stochastic differential equations with non-Gaussian L\'evy noise are suitable to model these systems. Thus it is desirable and essential to infer such equations from available data to reasonably predict dynamical behaviors. In this work, we consider a data-driven method to extract stochastic dynamical systems with non-Gaussian asymmetric (rather than the symmetric) L\'evy process, as well as Gaussian Brownian motion. We establish a theoretical framework and design a numerical algorithm to compute the asymmetric L\'evy jump measure, drift and diffusion (i.e., nonlocal Kramers-Moyal formulas), hence obtaining the stochastic governing law, from noisy data. Numerical experiments on several prototypical examples confirm the efficacy and accuracy of this method. This method will become an effective tool in discovering the governing laws from available data sets and in understanding the mechanisms underlying complex random phenomena.

Related papers

• Learning Controlled Stochastic Differential Equations [61.82896036131116]
  This work proposes a novel method for estimating both drift and diffusion coefficients of continuous, multidimensional, nonlinear controlled differential equations with non-uniform diffusion. We provide strong theoretical guarantees, including finite-sample bounds for (L2), (Linfty), and risk metrics, with learning rates adaptive to coefficients' regularity. Our method is available as an open-source Python library.
  arXiv  Detail & Related papers  (2024-11-04T11:09:58Z)
• Weak Collocation Regression for Inferring Stochastic Dynamics with L\'{e}vy Noise [8.15076267771005]
  We propose a weak form of the Fokker-Planck (FP) equation for extracting dynamics with L'evy noise. Our approach can simultaneously distinguish mixed noise types, even in multi-dimensional problems.
  arXiv  Detail & Related papers  (2024-03-13T06:54:38Z)
• Fusion of Gaussian Processes Predictions with Monte Carlo Sampling [61.31380086717422]
  In science and engineering, we often work with models designed for accurate prediction of variables of interest. Recognizing that these models are approximations of reality, it becomes desirable to apply multiple models to the same data and integrate their outcomes.
  arXiv  Detail & Related papers  (2024-03-03T04:21:21Z)
• HyperSINDy: Deep Generative Modeling of Nonlinear Stochastic Governing Equations [5.279268784803583]
  We introduce HyperSINDy, a framework for modeling dynamics via a deep generative model of sparse governing equations from data. Once trained, HyperSINDy generates dynamics via a differential equation whose coefficients are driven by a white noise. In experiments, HyperSINDy recovers ground truth governing equations, with learnedity scaling to match that of the data.
  arXiv  Detail & Related papers  (2023-10-07T14:41:59Z)
• Learning minimal representations of stochastic processes with variational autoencoders [52.99137594502433]
  We introduce an unsupervised machine learning approach to determine the minimal set of parameters required to describe a process. Our approach enables for the autonomous discovery of unknown parameters describing processes.
  arXiv  Detail & Related papers  (2023-07-21T14:25:06Z)
• Capturing dynamical correlations using implicit neural representations [85.66456606776552]
  We develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data. In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data.
  arXiv  Detail & Related papers  (2023-04-08T07:55:36Z)
• Score-based Diffusion Models in Function Space [140.792362459734]
  Diffusion models have recently emerged as a powerful framework for generative modeling. We introduce a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• A Robust and Flexible EM Algorithm for Mixtures of Elliptical Distributions with Missing Data [71.9573352891936]
  This paper tackles the problem of missing data imputation for noisy and non-Gaussian data. A new EM algorithm is investigated for mixtures of elliptical distributions with the property of handling potential missing data. Experimental results on synthetic data demonstrate that the proposed algorithm is robust to outliers and can be used with non-Gaussian data.
  arXiv  Detail & Related papers  (2022-01-28T10:01:37Z)
• Extracting stochastic dynamical systems with $\alpha$-stable L\'evy noise from data [14.230182518492311]
  We propose a data-driven method to extract systems with $alpha$-stable L'evy noise from short burst data. More specifically, we first estimate the L'evy jump measure and noise intensity. Then we approximate the drift coefficient by combining nonlocal Kramers-Moyal formulas with normalizing flows.
  arXiv  Detail & Related papers  (2021-09-30T06:57:42Z)
• Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas [3.8325907381729496]
  We propose a data-driven approach to extract governing laws with both (Gaussian) Brownian motion and (non-Gaussian) L'evy motion. We demonstrate that this approach can learn the differential equation with L'evy motion.
  arXiv  Detail & Related papers  (2021-08-28T04:56:51Z)
• A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise [5.17900889163564]
  We develop a new data-driven approach to extract governing laws from noisy data sets. First, we establish a feasible theoretical framework, by expressing the drift coefficient, diffusion coefficient and jump measure. We then design a numerical algorithm to compute the drift, diffusion coefficient and jump measure, and thus extract a governing equation with Gaussian and non-Gaussian noise.
  arXiv  Detail & Related papers  (2020-05-07T21:29:17Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.