Generalizing the SINDy approach with nested neural networks

TL;DR

Nested SINDy enhances the traditional SINDy framework by incorporating nested neural network structures, significantly improving its ability to identify complex symbolic expressions from data.

Contribution

The paper introduces Nested SINDy, a novel extension of SINDy that increases expressivity for symbolic regression of complex systems using nested neural network layers.

Findings

Nested SINDy accurately identifies symbolic expressions for simple systems.

It produces sparse, approximate representations for complex systems.

The method surpasses traditional SINDy in expressivity.

Abstract

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.