Reconstruction of delay differential equations via learning parameterized dictionary

Abstract

Reconstructing the governing equations of unknown systems plays an important role in physics. Sparse modeling methods, such as sparse identification of nonlinear dynamical systems (SINDy), provide a very good framework to tackle this problem. However, SINDy struggles with discovering delay differential equations (DDEs) due to the curse of dimensionality in the candidate dictionary. That is, each possible discrete delay term in the dictionary should be regarded as an independent basis element, which makes the combination of exhaustive discrete delay terms blow up. To deal with the curse of dimensionality and further reconstruct DDEs, a variant method of SINDy is presented in this paper in which a parameterized dictionary is proposed. In the parameterized dictionary, undetermined parameter variables are included in the candidate function to cover a wide range of different functions. Then, the reconstruction problem is reformulated as mixed-integer nonlinear programming (MINLP) and solved by a bi-level algorithm in which the binary version and the continuous version of particle swarm optimization (PSO) are respectively adopted in two levels. Simulation experiments are performed in 5 test systems including 3 well-known chaotic DDEs such as Mackey–Glass system. The results validate the effectiveness of the presented method.