An end-to-end deep learning approach for extracting stochastic dynamical systems with $α$-stable Lévy noise

Extracting stochastic dynamical systems with $α$-stable Lévy noise from data

With the rapid increase of valuable observational, experimental and simulated data for complex systems, much efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the wide applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extract stochastic dynamical systems with (non-Gaussian) Lévy noise are relatively few so far. In this work, we propose a data-driven method to extract stochastic dynamical systems with $α$-stable Lévy noise from short burst data based on the properties of $α$-stable dis…

Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas

With the rapid development of computational techniques and scientific tools, great progress of data-driven analysis has been made to extract governing laws of dynamical systems from data. Despite the wide occurrences of non-Gaussian fluctuations, the effective data-driven methods to identify stochastic differential equations with non-Gaussian Lévy noise are relatively few so far. In this work, we propose a data-driven approach to extract stochastic governing laws with both (Gaussian) Brownian motion and (non-Gaussian) Lévy motion, from short bursts of simulation data. Specifically, we use the …

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

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 Lévy 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 …

A Single-Timescale Method for Stochastic Bilevel Optimization

Stochastic bilevel optimization generalizes the classic stochastic optimization from the minimization of a single objective to the minimization of an objective function that depends the solution of another optimization problem. Recently, stochastic bilevel optimization is regaining popularity in emerging machine learning applications such as hyper-parameter optimization and model-agnostic meta learning. To solve this class of stochastic optimization problems, existing methods require either double-loop or two-timescale updates, which are sometimes less efficient. This paper develops a new opti…

Drift Identification for Lévy alpha-Stable Stochastic Systems

This paper focuses on a stochastic system identification problem: given time series observations of a stochastic differential equation (SDE) driven by Lévy $α$-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 se…

通过学习预测相关性从多个大规模神经群体记录中提取低维动力学

Extracting low-dimensional dynamics from multiple large-scale neural population recordings by learning to predict correlations

理解神经种群动力学的一个有效方法是使用降维方法从种群记录中提取低维轨迹。目前的神经数据降维方法仅限于单个种群记录，无法识别嵌入多个测量的动态。我们提出了一种从多个连续记录中提取低维动态的方法。我们的算法可以扩展到包含数百万个观测维度的数据，从而可以访问分布在大量人群或多个大脑区域的动态。基于动力系统的子空间识别方法，我们使用可扩展的随机梯度下降算法通过最小化时刻匹配目标来进行参数估计：该模型被优化以预测跨神经元和跨时间的时间协变。我们展示了这种方法如何自然地处理缺失数据和多个部分记录，并可以识别动态和预测相关性，即使在存在严重的二次采样和记录之间的小重叠。我们证明了模拟数据和全脑幼体斑马鱼成像数据集的方法的有效性。