Discovering Laws from Observations: A Data-Driven Approach

Abstract

A variety of complex patterns displayed by animal physiology, microbial communities, biological systems, or even artificial networks such as neural networks can be modeled by mathematical techniques which use non-linear, non-stationary, non-Markovian (i.e., long-range dependence) properties, to name a few. To identify the non-stationary changes over time, the models utilizing partial differential equations (PDEs) work well which track minute changes as well as the driving force. Further, the fractional PDEs have the flexibility of modeling long-range dependence across a sample trajectory. The scale-invariance in the magnitudes, as well as long-range dependence across time in a diffusion process, is captured by having fractional operators for both space and time. In this work, we propose to utilize the fractional PDEs to model sample trajectories and provide an estimation of the associated process with fewer samples. The space-time fractional diffusion process is generalized with the diffusion coefficient as well as drift (or advection) terms, which are domain-specific and tunable. Instead of usual methods to model dynamics of the system, the proposed techniques aim at modeling the minute changes in the dynamical system along with scale-invariance properties as well as long temporal dependence. With the essence of Dynamic Data-Driven Applications Systems (DDDAS), we let the data decide which model to use. We estimate all the parameters of the involved generalized fractional PDE by solving optimization problems minimizing error between empirical and theoretical fractional moments. To demonstrate the effectiveness of the proposed algorithm in retrieving the parameters, we perform an extensive set of simulations with various parameters’ combinations.