  Symbolic Regression (SR) is a widely studied field of research that aims to
infer symbolic expressions from data. A popular approach for SR is the Sparse
Identification of Nonlinear Dynamical Systems (\sindy) framework, which uses
sparse regression to identify governing equations from data. This study
introduces an enhanced method, Nested SINDy, that aims to increase the
expressivity of the SINDy approach thanks to a nested structure. Indeed,
traditional symbolic regression and system identification methods often fail
with complex systems that cannot be easily described analytically. Nested SINDy
builds on the SINDy framework by introducing additional layers before and after
the core SINDy layer. This allows the method to identify symbolic
representations for a wider range of systems, including those with compositions
and products of functions. We demonstrate the ability of the Nested SINDy
approach to accurately find symbolic expressions for simple systems, such as
basic trigonometric functions, and sparse (false but accurate) analytical
representations for more complex systems. Our results highlight Nested SINDy's
potential as a tool for symbolic regression, surpassing the traditional SINDy
approach in terms of expressivity. However, we also note the challenges in the
optimization process for Nested SINDy and suggest future research directions,
including the designing of a more robust methodology for the optimization
process. This study proves that Nested SINDy can effectively discover symbolic
representations of dynamical systems from data, offering new opportunities for
understanding complex systems through data-driven methods.
