Identifying Arguments of Space-Time Fractional Diffusion: Data-Driven Approach - Znaidi, Mohamed Ridha; Gupta, Gaurav; Asgari, Kamiar; Bogdan, Paul (Frontiers in Applied Mathematics and Statistics, 2020)
A plethora of dynamical complex systems, from disordered media to living systems, exhibit anomalous diffusion behavior that is characterized by a nonlinear mean-square displacement relationship. From a mathematical perspective, the anomalous behavior is described by a partial differential equation (PDE) model that often involves fractional derivatives. In general, PDEs represent various physical laws/rules governing complex dynamical systems. Since prior knowledge about the physical laws is not available, in the diverse areas of engineering involving complex systems, we aim to discover the PDE and its parameters from system's evolution data. Towards deciphering non-trivial mechanisms driving a complex system, we propose a data-driven approach that estimates the parameters of a PDE model associated with the anomalous diffusion. We study the space-time fractional diffusion model that describes a complex stochastic process, where the magnitude and the time increments are stable processes. Starting from time-series data recorded while the system is evolving, we develop a higher order moments-based approach to determine the parameters of a generalized fractional PDE. We formulate a regression problem to allow us to estimate the arguments of the fractional diffusion. Finally, using extensive simulation studies, we show that the proposed approach is effective at retrieving the relevant parameters of the space-time fractional PDE.