2 research outputs found
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