Computing Lyapunov functions using deep neural networks

2020/05/18 by Grüne, Lars · 4 citations
#34D20 #65Y20 #93D30 #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2005.08965

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.

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• Physics-Informed Machine Learning for Characterizing System Stability
• Lyapunov Stability Learning with Nonlinear Control via Inductive Biases
• Convexity and strict convexity for compositional neural networks in high-dimensional optimal control
• Strong Lyapunov functions for rough systems

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