Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation

TOOL LyZNet: A Lightweight Python Tool for Learning and Verifying Neural Lyapunov Functions and Regions of Attraction

“…Many authors have recently investigated the use of neural networks for computing Lyapunov functions (see, e.g., [2,4,12,12,17,22], and [9] for a recent survey). In fact, such efforts date back to as early as the 1990s [30,40].…”

“…Physics-informed neural solution to Zubov's PDE. Put together, ( 9), (12), and (13) allow us to learn a neural Lyapunov function via physics-informed neural networks [25,41] for solving PDEs.…”

“…evaluated over a set of collocation points {𝑥 𝑖 } 𝑁 𝑖=1 chosen over a domain 𝑋 . For instance, we can choose Ψ as in (12). The loss L boundary captures boundary conditions.…”

“…Example 5.4 (10-dimensional system). The example was used by the authors of [16] and [12] to illustrate the training of neural network Lyapunov functions for stability analysis, where the trained Lyapunov functions were not verified. Here, we use the example to illustrate the compositional verification functionalities of LyZNet.…”

Discovering efficient periodic behaviors in mechanical systems via neural approximators

“…In particular, data-driven methods, for example, neural networks, have often been used as function approximators for learning the dynamics of systems [14][15][16][17] or for representing control strategies [18][19][20] even in high-dimensional optimal control problems. 21 Furthermore, neural networks can be used to approximate Lyapunov functions in the case of autonomous [22][23][24][25][26] and nonautonomous dynamical systems [27][28][29][30] for stability and control purposes. However, purely data-driven methods often learn physically inconsistent models that do not respect physical conservation laws.…”