Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913871360425984 |
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| author | Liu, Jun Meng, Yiming Fitzsimmons, Maxwell Zhou, Ruikun |
| author_facet | Liu, Jun Meng, Yiming Fitzsimmons, Maxwell Zhou, Ruikun |
| contents | We provide a systematic investigation of using physics-informed neural networks to compute Lyapunov functions. We encode Lyapunov conditions as a partial differential equation (PDE) and use this for training neural network Lyapunov functions. We analyze the analytical properties of the solutions to the Lyapunov and Zubov PDEs. In particular, we show that employing the Zubov equation in training neural Lyapunov functions can lead to approximate regions of attraction close to the true domain of attraction. We also examine approximation errors and the convergence of neural approximations to the unique solution of Zubov's equation. We then provide sufficient conditions for the learned neural Lyapunov functions that can be readily verified by satisfiability modulo theories (SMT) solvers, enabling formal verification of both local stability analysis and region-of-attraction estimates in the large. Through a number of nonlinear examples, ranging from low to high dimensions, we demonstrate that the proposed framework can outperform traditional sums-of-squares (SOS) Lyapunov functions obtained using semidefinite programming (SDP). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09131 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification Liu, Jun Meng, Yiming Fitzsimmons, Maxwell Zhou, Ruikun Optimization and Control Machine Learning Systems and Control We provide a systematic investigation of using physics-informed neural networks to compute Lyapunov functions. We encode Lyapunov conditions as a partial differential equation (PDE) and use this for training neural network Lyapunov functions. We analyze the analytical properties of the solutions to the Lyapunov and Zubov PDEs. In particular, we show that employing the Zubov equation in training neural Lyapunov functions can lead to approximate regions of attraction close to the true domain of attraction. We also examine approximation errors and the convergence of neural approximations to the unique solution of Zubov's equation. We then provide sufficient conditions for the learned neural Lyapunov functions that can be readily verified by satisfiability modulo theories (SMT) solvers, enabling formal verification of both local stability analysis and region-of-attraction estimates in the large. Through a number of nonlinear examples, ranging from low to high dimensions, we demonstrate that the proposed framework can outperform traditional sums-of-squares (SOS) Lyapunov functions obtained using semidefinite programming (SDP). |
| title | Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification |
| topic | Optimization and Control Machine Learning Systems and Control |
| url | https://arxiv.org/abs/2312.09131 |