Koopman operator for time-dependent reliability analysis

TL;DR

This paper introduces a Koopman operator-based deep learning method for time-dependent reliability analysis of nonlinear dynamical systems, capable of handling uncertainties and outperforming traditional neural networks.

Contribution

It develops an end-to-end deep learning architecture to identify Koopman observables for robust, out-of-distribution time-dependent reliability analysis in nonlinear systems.

Findings

Outperforms auto-regressive neural networks and LSTM in numerical examples.

Robust to uncertainties in initial conditions and system parameters.

Generalizes well to unseen environments.

Abstract

Time-dependent structural reliability analysis of nonlinear dynamical systems is non-trivial; subsequently, scope of most of the structural reliability analysis methods is limited to time-independent reliability analysis only. In this work, we propose a Koopman operator based approach for time-dependent reliability analysis of nonlinear dynamical systems. Since the Koopman representations can transform any nonlinear dynamical system into a linear dynamical system, the time evolution of dynamical systems can be obtained by Koopman operators seamlessly regardless of the nonlinear or chaotic behavior. Despite the fact that the Koopman theory has been in vogue a long time back, identifying intrinsic coordinates is a challenging task; to address this, we propose an end-to-end deep learning architecture that learns the Koopman observables and then use it for time marching the dynamical response. Unlike purely data-driven approaches, the proposed approach is robust even in the presence of uncertainties; this renders the proposed approach suitable for time-dependent reliability analysis. We propose two architectures; one suitable for time-dependent reliability analysis when the system is subjected to random initial condition and the other suitable when the underlying system have uncertainties in system parameters. The proposed approach is robust and generalizes to unseen environment (out-of-distribution prediction). Efficacy of the proposed approached is illustrated using three numerical examples. Results obtained indicate supremacy of the proposed approach as compared to purely data-driven auto-regressive neural network and long-short term memory network.