Combining Neural Networks and Symbolic Regression for Analytical Lyapunov Function Discovery

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.