Extracting Non-Gaussian Governing Laws from Data on Mean Exit Time

Motivated by the existing difficulties in establishing mathematical models and in observing the system state time series for some complex systems, especially for those driven by non-Gaussian Levy motion, we devise a method for extracting non-Gaussian governing laws with observations only on mean exit time. It is feasible to observe mean exit time for certain complex systems. With the observations, a sparse regression technique in the least squares sense is utilized to obtain the approximated function expression of mean exit time. Then, we learn the generator and further identify the stochastic differential equations through solving an inverse problem for a nonlocal partial differential equation and minimizing an error objective function. Finally, we verify the efficacy of the proposed method by three examples with the aid of the simulated data from the original systems. Results show that the method can apply to not only the stochastic dynamical systems driven by Gaussian Brownian motion but also those driven by non-Gaussian Levy motion, including those systems with complex rational drift

A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise

With the rapid increase of valuable observational, experimental and simulating data for complex systems, great efforts are being devoted to discovering governing laws underlying the evolution of these systems. However, the existing techniques are limited to extract governing laws from data as either deterministic differential equations or stochastic differential equations with Gaussian noise. In the present work, we develop a new data-driven approach to extract stochastic dynamical systems with non-Gaussian symmetric L\'evy noise, as well as Gaussian noise. First, we establish a feasible theoretical framework, by expressing the drift coefficient, diffusion coefficient and jump measure (i.e., anomalous diffusion) for the underlying stochastic dynamical system in terms of sample paths data. We then design a numerical algorithm to compute the drift, diffusion coefficient and jump measure, and thus extract a governing stochastic differential equation with Gaussian and non-Gaussian noise. Finally, we demonstrate the efficacy and accuracy of our approach by applying to several prototypical one-, two- and three-dimensional systems. This new approach will become a tool in discovering governing dynamical laws from noisy data sets, from observing or simulating complex phenomena, such as rare events triggered by random fluctuations with heavy as well as light tail statistical features.Comment: 36 page