(Un)supervised Learning of Maximal Lyapunov Functions
This addresses the challenge of analyzing stability in dynamical systems, which is crucial for control and robotics applications, representing a novel method rather than an incremental improvement.
The paper tackles the problem of discovering maximal Lyapunov functions to determine the region of attraction for dynamical systems, proposing a novel neural network architecture that is proven to be a universal approximator and matches or outperforms state-of-the-art methods in accuracy.
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.