Analytical Lyapunov Function Discovery: An RL-based Generative Approach
Authors: Haohan Zou, Jie Feng, Hao Zhao, Yuanyuan Shi
ICML 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We demonstrate the efficiency of our approach on a range of nonlinear dynamical systems with up to ten dimensions and show that it can discover Lyapunov functions not previously identified in the control literature. Full implementation is available on Github. ... We validate the proposed algorithm across a variety of nonlinear dynamics by finding their local Lyapunov functions at the equilibrium point to verify their stability... Table 1 summarizes the runtime, success rate, and discovered Lyapunov functions for a selection of tested nonlinear systems, ranging from 2-D to 10-D, demonstrating the robustness and scalability of our framework. |
| Researcher Affiliation | Academia | 1Department of Electrical and Computer Engineering, UC San Diego, La Jolla, United States 2Swiss Federal Institute of Technology in Lausanne. Correspondence to: Haohan Zou <EMAIL>. |
| Pseudocode | Yes | Algorithm 1 Training Framework for Analytical Lyapunov Function Discovery via Reinforcement Learning Input: Dynamics f(x), state space D, quantile α, symbolic library L, batch size Q, max complexity k, radius r. Output: Valid Lyapunov function V for system f(x). ... Algorithm 2 Global-optimization-based Numerical Verification Input: A set of analytical expressions V = {V i| i = 1, , Q}, radius r, and state space D. Output: a set of numerically valid candidate V , a set of encountered counterexample Xce. ... Algorithm 3 Expert Guidance Loss Input: Elite set of analytical expressions Vgp, input dynamics f(x), and transformer parameters ϕ. Output: The weighted cross-entropy loss between the transformer model output probability distribution and given refined expressions Vgp. |
| Open Source Code | Yes | Full implementation is available on Github. |
| Open Datasets | No | We categorize our collected test dynamical systems into two kinds: 1). Polynomial Systems, and 2). Non-polynomial Systems, where three polynomial systems are adopted from Alfarano et al. (2024) (Appendices F.2 & F.3), and others come from real-world examples. Detailed information about systems is summarized in Appendices F and G. |
| Dataset Splits | No | Randomly sample training datapoints X = {x1, , x N} where xi D, ... In datapoint sampling for counter-example identification, for each candidate expression, 800 data points are sampled from each of Br(x 1) and Br(x 2), and additional 800 data points are randomly sampled across the state space D. |
| Hardware Specification | Yes | Computation Resources: All experiments in this work were performed on a workstation with an AMD Ryzen Threadripper 2920X 12-Core Processor, an Nvidia RTX 2080Ti GPU 11 GB, and an Nvidia RTX 2080 GPU 8 GB. |
| Software Dependencies | No | We use d Real Gao et al. (2013) SMT solver for final verification of found Lyapunov functions... We compare our proposed framework against four baseline algorithms... SOS methods via SOSTOOLS (Matlab) (Papachristodoulou et al., 2013)... we incorporate a Genetic Programming (GP) component (DEAP Fortin et al. (2012)) into the training paradigm. |
| Experiment Setup | Yes | Implementation Details of Framework. Detailed explanation for all components in our framework is presented in Appendices A (Transformer), B (Global-optimization-based Numerical Verification), C (Risk-seeking Policy Gradient), and D (Genetic Programming). The symbolic library Ls is defined as {+, , , sin, cos, xi} in all tests. ... In this work, we set the embedding dimension to 128, attention head to 2, and applied a 6-layer transformer decoder for the candidate expression generation. In each epoch, we sample Q = 500 expressions as candidates. ... In implementation, we choose α = 0.1 in the training for all tested dynamics. |