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.

Symbolic Regression（记号回归）是一个广泛研究的领域，旨在通过数据推断出记号表达式。本研究介绍了一种增强方法，名为Nested SINDy，旨在通过引入嵌套结构来增加SINDy方法的表达能力。Nested SINDy通过在核心SINDy层之前和之后引入附加层，能够对更广泛的系统进行符号表示。我们的研究结果突出了Nested SINDy在符号回归方面的潜力，超越传统的SINDy方法在表达能力上。然而，我们也注意到了Nested SINDy在优化过程中的挑战，并建议未来的研究方向，包括设计更稳健的优化方法论。本研究证明Nested SINDy能够有效地从数据中发现动态系统的符号表示，为通过数据驱动方法理解复杂系统提供了新机遇。

利用嵌套神经网络推广SINDy方法