Data-driven discovery of Caputo fractional order systems

Abstract

Due to the peculiar non-locality of fractional order differential and integral operators, Caputo fractional order systems are harder than integer order systems to be discoverd from data. To solve this open problem, we propose a framework of method capable of discovering Caputo fractional order (autonomous and non-autonomous) systems from measurable data. The interior point method and genetic algorithm are embedded respectively in the framework. The former is mainly presented in text, while the latter is implemented for comparison and validation. The framework is designed to dynamically and coordinately update the fractional order and vector field function for the system to be discovered till the difference between the measured and discovered systems is minimized. It is computationally efficient, robust and illustrated by discovering the Caputo fractional order Lorenz system, Chua’s circuit and Duffing’s oscillator hidden in measured data. As thus, this work provides one way to uncover underlying Caputo fractional order mathematical models (or physical laws and governing equations).