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Computing Lyapunov functions using deep neural networks

Title data

Grüne, Lars:
Computing Lyapunov functions using deep neural networks.
Bayreuth , 2020 . - 27 p.

Official URL: Volltext

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Abstract in another language

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.

Further data

Item Type: Preprint, postprint
Keywords: deep neural network; Lyapunov function; stability; small-gain condition; curse of dimensionality; training algorithm
Subject classification: Mathematics Subject Classification Code: 93D30, 34D20, 65Y20
Institutions of the University: Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) > Chair Mathematics V (Applied Mathematics) - Univ.-Prof. Dr. Lars Grüne
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Applied Mathematics
Profile Fields
Profile Fields > Advanced Fields
Profile Fields > Advanced Fields > Nonlinear Dynamics
Research Institutions > Research Centres > Forschungszentrum für Modellbildung und Simulation (MODUS)
Research Institutions
Research Institutions > Research Centres
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 20 May 2020 08:51
Last Modified: 20 May 2020 08:51
URI: https://eref.uni-bayreuth.de/id/eprint/55192