Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weak Collocation Regression for Inferring Stochastic Dynamics with Lévy Noise

Liya Guo, Liwei Lu, Zhijun Zeng, Pipi Hu, Yi Zhu

Abstract

With the rapid increase of observational, experimental and simulated data for stochastic systems, tremendous efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the broad applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extracting stochastic dynamics with Lévy noise are relatively few. In this work, we propose a Weak Collocation Regression (WCR) to explicitly reveal unknown stochastic dynamical systems, i.e., the Stochastic Differential Equation (SDE) with both $α$-stable Lévy noise and Gaussian noise, from discrete aggregate data. This method utilizes the evolution equation of the probability distribution function, i.e., the Fokker-Planck (FP) equation. With the weak form of the FP equation, the WCR constructs a linear system of unknown parameters where all integrals are evaluated by Monte Carlo method with the observations. Then, the unknown parameters are obtained by a sparse linear regression. For a SDE with Lévy noise, the corresponding FP equation is a partial integro-differential equation (PIDE), which contains nonlocal terms, and is difficult to deal with. The weak form can avoid complicated multiple integrals. Our approach can simultaneously distinguish mixed noise types, even in multi-dimensional problems. Numerical experiments demonstrate that our method is accurate and computationally efficient.

Weak Collocation Regression for Inferring Stochastic Dynamics with Lévy Noise

Abstract

With the rapid increase of observational, experimental and simulated data for stochastic systems, tremendous efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the broad applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extracting stochastic dynamics with Lévy noise are relatively few. In this work, we propose a Weak Collocation Regression (WCR) to explicitly reveal unknown stochastic dynamical systems, i.e., the Stochastic Differential Equation (SDE) with both -stable Lévy noise and Gaussian noise, from discrete aggregate data. This method utilizes the evolution equation of the probability distribution function, i.e., the Fokker-Planck (FP) equation. With the weak form of the FP equation, the WCR constructs a linear system of unknown parameters where all integrals are evaluated by Monte Carlo method with the observations. Then, the unknown parameters are obtained by a sparse linear regression. For a SDE with Lévy noise, the corresponding FP equation is a partial integro-differential equation (PIDE), which contains nonlocal terms, and is difficult to deal with. The weak form can avoid complicated multiple integrals. Our approach can simultaneously distinguish mixed noise types, even in multi-dimensional problems. Numerical experiments demonstrate that our method is accurate and computationally efficient.
Paper Structure (37 sections, 1 theorem, 36 equations, 5 figures, 10 tables, 2 algorithms)

This paper contains 37 sections, 1 theorem, 36 equations, 5 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. SDE with Lévy Noise and Its Fokker-Planck Equation
  3. Methodology
  4. Overview
  5. Approximate the Weak Form of FP equation by Monte Carlo Method
  6. Weak formula.
  7. Temporal derivative and Monte Carlo approximation.
  8. Build Linear Regression Equations
  9. Selected basis.
  10. Assembly of block linear equations.
  11. Solution of unknown parameters.
  12. Experimental Results
  13. Experimental Setting
  14. Data.
  15. Basis and kernel functions.
  16. ...and 22 more sections

Key Result

Lemma 1

The fractional Laplacian operator has translation and scaling properties. For Schwartz functions $\psi(\bm{x})$, $\bm{x} \in \mathbb{R}^d$ with rapid descent properties, let $\Psi(x) = (-\Delta)^{\alpha/2} \psi(\bm{x})$, and for $\alpha > 0$, the followings hold. Specially, if $\psi(\bm{x})$ is in the form of $e^{-\left| \bm{x} \right|^2}$, then for $\alpha \in(0, 1) \cup (1, 2)$, the fractional

Figures (5)

  • Figure 1: Framework of our method. Take the case of independent dimensions as an example. We start by discretely observing the particle motion in a stochastic dynamic system, capturing $L$ snapshots and using the observed particle positions as data set $\mathbb{X}$ (part (i)). Assume that the stochastic system satisfies a SDE with Lévy noise (part (ii)). Specifically, the WCR involves the following five steps: (a) Select Gaussian kernel functions $\psi(\bm{x}, \rho_m, \gamma)$ with the same variance $\gamma$ and utilize the weak form of the Fokker-Planck equation to act the derivatives to the known kernel functions. (b) Approximate the weak formula using aggregate data by the Monte Carlo method and temporal difference. (c) Select basis and expand coefficient functions as combinations of basis, then obtain a linear equation system; (d) Construct a linear equation block $\bm{A}_m \bm{\zeta} = \bm{\hat{H}}_m$ for each sampled Gaussian kernel. They are assembled to a linear equation system of multiple blocks $\bm{\tilde{A}} \bm{\zeta} = \bm{\tilde{H}}$. Solve it by sparse regression; (e) Estimate the coefficients of the basis expansion by regression.
  • Figure 2: Comparison of the distribution of true data and the predicted distributions of estimated SDE with or without Lévy noise. The top row represents results obtained from WCR only with Gaussian noise and the bottom row shows the predicted distribution by WCR method based on both Gaussian and Lévy noise. The data distributions are plotted at $t=0.75$ (left) and $t=1.2$ (right).
  • Figure 3: Comparison of the true data distribution and the predicted distribution of the estimated SDE at the end moment $t=1.2$ for $3d$ coupled system. The "WD" in figures means "Wasserstein Distance" of the predicted one and the ground truth.
  • Figure 4: Comparison of the true data distribution and the predicted distribution of the estimated SDE at moments $t=1.2, 1.4, 1.8$ for $5d$ coupled system. The "WD" in sub-titles means "Wasserstein Distance" of the predicted one and the ground truth.
  • Figure 5: The estimated drift terms of SDE with non-polynomial drift term $m(x) = - 4x^3-2x e^{-x^2}$.

Theorems & Definitions (6)

  • Lemma 1
  • Remark 1
  • Example 1
  • Example 2
  • Example 3: Geometric Brownian motion
  • Example 4