# (Un)supervised Learning of Maximal Lyapunov Functions

arXiv: https://arxiv.org/abs/2408.17246 · HTML: https://mlsanity.com/paper/2408.17246/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 54 references that are in the map.
- First posted on arXiv 30 Aug 2024.

## Builds on (most cited first, 21 of 21)

- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems](https://mlsanity.com/paper/1808.00924/) (2018)
- [Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling](https://mlsanity.com/paper/2207.02338/) (2022)
- [Lyapunov-stable neural-network control](https://mlsanity.com/paper/2109.14152/) (2021)
- [Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees](https://mlsanity.com/paper/2206.01913/) (2022)
- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification](https://mlsanity.com/paper/2312.09131/) (2023)
- [Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation](https://mlsanity.com/paper/2109.13359/) (2021)
- [Computing Lyapunov functions using deep neural networks](https://mlsanity.com/paper/2005.08965/) (2020)
- [Physics-informed Learning for Identification and State Reconstruction of Traffic Density](https://mlsanity.com/paper/2103.13852/) (2021)
- [Actor-Critic Physics-informed Neural Lyapunov Control](https://mlsanity.com/paper/2403.08448/) (2024)
- [Neural Lyapunov Differentiable Predictive Control](https://mlsanity.com/paper/2205.10728/) (2022)
- [A Curriculum-Training-Based Strategy for Distributing Collocation Points during Physics-Informed Neural Network Training](https://mlsanity.com/paper/2211.11396/) (2022)
- [Region of Attraction Estimation Using Invariant Sets and Rational Lyapunov Functions](https://mlsanity.com/paper/1604.01084/) (2016)
- [Converse Lyapunov Functions and Converging Inner Approximations to Maximal Regions of Attraction of Nonlinear Systems](https://mlsanity.com/paper/2103.12825/) (2021)

## Abstract (arXiv)

In this paper, we address the problem of discovering maximal Lyapunov functions, as a means of determining the region of attraction of a dynamical system. To this end, we design a novel neural network architecture, which we prove to be a universal approximator of (maximal) Lyapunov functions. The architecture combines a local quadratic approximation with the output of a neural network, which models global higher-order terms in the Taylor expansion. We formulate the problem of training the Lyapunov function as an unsupervised optimization problem with dynamical constraints, which can be solved leveraging techniques from physics-informed learning. We propose and analyze a tailored training algorithm, based on the primal-dual algorithm, that can efficiently solve the problem. Additionally, we show how the learning problem formulation can be adapted to integrate data, when available. We apply the proposed approach to different classes of systems, showing that it matches or outperforms state-of-the-art alternatives in the accuracy of the approximated regions of attraction.