  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.
