Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation
This addresses the challenge of stability analysis in nonlinear control for researchers and engineers, offering a more efficient and scalable solution, though it appears incremental as it builds on existing neural network approaches with specific architectural enhancements.
The paper tackles the problem of approximating Lyapunov functions for high-dimensional dynamical systems by developing Lyapunov-Net, a deep neural network architecture that guarantees positive definiteness and simplifies training with a single-term empirical risk, resulting in significant performance improvements over state-of-the-art methods on systems up to 30 dimensions.
We develop a versatile deep neural network architecture, called Lyapunov-Net, to approximate Lyapunov functions of dynamical systems in high dimensions. Lyapunov-Net guarantees positive definiteness, and thus it can be easily trained to satisfy the negative orbital derivative condition, which only renders a single term in the empirical risk function in practice. This significantly reduces the number of hyper-parameters compared to existing methods. We also provide theoretical justifications on the approximation power of Lyapunov-Net and its complexity bounds. We demonstrate the efficiency of the proposed method on nonlinear dynamical systems involving up to 30-dimensional state spaces, and show that the proposed approach significantly outperforms the state-of-the-art methods.