Analytical-Lyapunov-Function-Discovery

Offical Pytorch implementation for the paper "Analytical Lyapunov Function Discovery: An RL-based Generative Approach", presented at ICML 2025. We introduce the first RL-based framework for directly discovering analytical Lyapunov functions for nonlinear dynamical systems, bypassing the need for supervised learning with large-scale datasets. Our framework succeeds on various non-polynomial dynamics, like the simple pendulum, quadrotor, and power system frequency control, and notably scales to a 10-D system and discovers a valid local Lyapunov function for power system frequency control with lossy transmission lines, which is previously unknown in the literature. For details, see Analytical Lyapunov Function Discovery: An RL-based Generative Approach.

Installation

Clone this repository and install the required Conda environment and dependencies by running:

run install.sh

Input Test Dynamics and Training Parameters

Before training, you need to specify the test dynamics and parameters defining the desirable output expressions. Follow these steps:

Define the Symbolic ODEs

In config.py, edit the function dynamics() to define your system's ODEs. Specify the state-space variables and return the dynamics in symbolic form.

Specify the State Space for Local Stability Analysis

Modify the state space domain $\mathcal{D}$ and the size of training set $\mathcal{X}$ in libs/sd3/dso/dso/task/regression/benchmarks_bkup_1.csv. Assign the corresponding entry name (e.g., fn_d_all_x) to the variable conf.exp.benchmark in Lyapunov_test_dso.py at Line 67.

Set Training Parameters

In config.py, update the config_factory() function to customize training hyperparameters, such as:

• symbolic library $\mathcal{L}$,

• length of sampled candidates (max & min),

• learning rates of training,

• hyperparameter $\epsilon$ of risk-seeking policy gradient and so on.

Note: In folder example_config, you can find a few examples on how to define system dynamics and relevant training parameters.

Training

Once your dynamics and training domain are configured, start training by running:

python Lyapunov_test_dso.py

Intermediate training statistics and final training result (if training converges) can be found in the log directory ./log/{$RUN}.

To configure and test the experiments with different initialization, please modify at the top of Lyapunov_test_dso.py

conf.exp.seed_start = 13

Citation

If you found this work useful or interesting for your own research, we would appreciate if you could cite our work:

@article{zou2025analytical,
  title={Analytical Lyapunov Function Discovery: An RL-based Generative Approach},
  author={Zou, Haohan and Feng, Jie and Zhao, Hao and Shi, Yuanyuan},
  journal={arXiv preprint arXiv:2502.02014},
  year={2025}
}

Feel free to leave any questions in the issues of Github or email the author at hazou@ucsd.edu.