We propose CoNSAL (Combining Neural networks and Symbolic regression for Analytical Lyapunov function) to construct analytical Lyapunov functions for nonlinear dynamic systems. This framework contains a neural Lyapunov function and a symbolic regression component, where symbolic regression is applied to distill the neural network to precise analytical forms. Our approach utilizes symbolic regression not only as a tool for translation but also as a means to uncover counterexamples. This procedure terminates when no counterexamples are found in the analytical formulation. Compared with previous results, CoNSAL directly produces an analytical form of the Lyapunov function with improved interpretability in both the learning process and the final results. We apply CoNSAL to 2-D inverted pendulum, path following, Van Der Pol Oscillator, 3-D trig dynamics, 4-D rotating wheel pendulum, 6-D 3-bus power system, and demonstrate that our algorithm successfully finds their valid Lyapunov functions. Code examples are available at https://github.com/HaohanZou/CoNSAL.

翻译：我们提出CoNSAL（结合神经网络与符号回归的解析李雅普诺夫函数）来为非线性动态系统构建解析李雅普诺夫函数。该框架包含一个神经李雅普诺夫函数和一个符号回归组件，其中符号回归用于将神经网络提炼为精确的解析形式。我们的方法不仅将符号回归作为转换工具，还将其用作发现反例的手段。当在解析表述中未发现反例时，该过程终止。与先前结果相比，CoNSAL直接生成李雅普诺夫函数的解析形式，在学习过程和最终结果方面均提升了可解释性。我们将CoNSAL应用于二维倒立摆、路径跟踪、范德波尔振荡器、三维三角动力学、四维旋转轮摆、六维三母线电力系统，并证明我们的算法成功找到了它们的有效李雅普诺夫函数。代码示例可在 https://github.com/HaohanZou/CoNSAL 获取。