Computing Lyapunov functions using deep neural networks
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Publication:6340960
DOI10.3934/JCD.2021006zbMATH Open1507.68267arXiv2005.08965MaRDI QIDQ6340960
Publication date: 18 May 2020
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.
Artificial neural networks and deep learning (68T07) Stability of solutions to ordinary differential equations (34D20)
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