# Drift Identification for L'evy alpha-Stable Stochastic Systems

arXiv: https://arxiv.org/abs/2212.03317 · HTML: https://mlsanity.com/paper/2212.03317/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 43 references that are in the map.
- First posted on arXiv 6 Dec 2022.

## Builds on (most cited first, 9 of 9)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019)
- [Fast Two-Sample Testing with Analytic Representations of Probability Measures](https://mlsanity.com/paper/1506.04725/) (2015)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise](https://mlsanity.com/paper/2005.03769/) (2020)
- [Extracting stochastic dynamical systems with α-stable L'evy noise from data](https://mlsanity.com/paper/2109.14881/) (2021)
- [Extracting Governing Laws from Sample Path Data of Non-Gaussian Stochastic Dynamical Systems](https://mlsanity.com/paper/2107.10127/) (2021)
- [An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable L'evy noise](https://mlsanity.com/paper/2201.13114/) (2022)
- [Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas](https://mlsanity.com/paper/2108.12570/) (2021)

## Built on by (most cited first, 1 of 1)

- [Weak Collocation Regression for Inferring Stochastic Dynamics with L'evy Noise](https://mlsanity.com/paper/2403.08292/) (2024)

## Abstract (arXiv)

This paper focuses on a stochastic system identification problem: given time series observations of a stochastic differential equation (SDE) driven by L'evy α-stable noise, estimate the SDE's drift field. For α in the interval [1,2), the noise is heavy-tailed, leading to computational difficulties for methods that compute transition densities and/or likelihoods in physical space. We propose a Fourier space approach that centers on computing time-dependent characteristic functions, i.e., Fourier transforms of time-dependent densities. Parameterizing the unknown drift field using Fourier series, we formulate a loss consisting of the squared error between predicted and empirical characteristic functions. We minimize this loss with gradients computed via the adjoint method. For a variety of one- and two-dimensional problems, we demonstrate that this method is capable of learning drift fields in qualitative and/or quantitative agreement with ground truth fields.