Reconstruction of Delay Differential Equation via Learning Parameterized Dictionary

Abstract

This paper presents a variant of sparse representation modeling method, which has a promising performance of reconstruction of delay differential equation from sampling data. In the new method, a parameterized dictionary of candidate functions is constructed against the traditional expanded dictionary. The parameterized dictionary uses a function with variables to represent a series of functions. It accordingly has the ability to express functions in the continuous function space so that the dimension of the dictionary can be exponentially decreased. This is particularly important when an exhaustion of candidate functions is needed to construct appropriate dictionary. The reconstruction of delay differential equation is such the case that each possible delay item should be considered as the basis to construct the dictionary and this naturally induces the curse of dimensionality. Correspondingly, the parameterized dictionary uses a variable to model the delay item so the curse disappears. Based on the parameterized dictionary, the reconstruction problem is then rewritten and treated as a mixed-integer nonlinear programming with both binary and continuous variables. To the best of our knowledge, such optimization problem is hard to solve with the traditional mathematical methods while the emerging evolutionary computation provides competitive solutions. Hence, the evolutionary computation technique is considered and a typical algorithm named particle swarm optimization is adopted in this paper. Experiments are carried out in 5 test systems including 3 well-known chaotic delay differential equations such as Mackey-Glass system. The experiment result shows the effectiveness of the new method to reconstruct delay differential equation.