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.

翻译：符号回归（SR）是一个广泛研究领域，旨在从数据中推断符号表达式。稀疏非线性动力学系统识别（SINDy）框架作为SR的常用方法，通过稀疏回归从数据中识别控制方程。本研究提出一种增强方法——嵌套SINDy，通过引入嵌套结构提升SINDy方法的表达能力。传统符号回归和系统辨识方法在处理难以解析描述的复杂系统时往往失效。嵌套SINDy在SINDy框架基础上，于核心SINDy层前后增加额外层，使该方法能够为更广泛的系统（包括含函数复合与乘积的系统）识别符号表示。我们证明了嵌套SINDy方法对简单系统（如基本三角函数）能精准找到符号表达式，对复杂系统则能发现稀疏（虽非精确但有效的）解析表示。结果表明嵌套SINDy在表达能力上超越传统SINDy方法，展现了作为符号回归工具的潜力。同时，本文指出嵌套SINDy优化过程中存在的挑战，并建议未来研究方向，包括设计更稳健的优化方法。本研究证实嵌套SINDy能从数据中有效发现动力系统的符号表示，为通过数据驱动方法理解复杂系统提供了新途径。