  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.
