Computing Lyapunov functions using deep neural networks

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Grüne, Lars:
Computing Lyapunov functions using deep neural networks.
In: Journal of Computational Dynamics. Bd. 8 (2021) Heft 2 . - S. 131-152.
ISSN 2158-2491
DOI: https://doi.org/10.3934/jcd.2021006

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Abstract

We propose a deep neural network architecture and a training algorithm for computing approximate Lyapunov functions of systems of nonlinear ordinary differential equations. Under the assumption that the system admits a compositional Lyapunov function, we prove that the number of neurons needed for an approximation of a Lyapunov function with fixed accuracy grows only polynomially in the state dimension, i.e., the proposed approach is able to overcome the curse of dimensionality. We show that nonlinear systems satisfying a small-gain condition admit compositional Lyapunov functions. Numerical examples in up to ten space dimensions illustrate the performance of the training scheme.