Koopman operator for time-dependent reliability analysis

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.

Surrogate assisted active subspace and active subspace assisted surrogate -- A new paradigm for high dimensional structural reliability analysis

Performing reliability analysis on complex systems is often computationally expensive. In particular, when dealing with systems having high input dimensionality, reliability estimation becomes a daunting task. A popular approach to overcome the problem associated with time-consuming and expensive evaluations is building a surrogate model. However, these computationally efficient models often suffer from the curse of dimensionality. Hence, training a surrogate model for high-dimensional problems is not straightforward. Henceforth, this paper presents a framework for solving high-dimensional reliability analysis problems. The basic premise is to train the surrogate model on a low-dimensional manifold, discovered using the active subspace algorithm. However, learning the low-dimensional manifold using active subspace is non-trivial as it requires information on the gradient of the response variable. To address this issue, we propose using sparse learning algorithms in conjunction with the active subspace algorithm; the resulting algorithm is referred to as the sparse active subspace (SAS) algorithm. We project the high-dimensional inputs onto the identified low-dimensional manifold identified using SAS. A high-fidelity surrogate model is used to map the inputs on the low-dimensional manifolds to the output response. We illustrate the efficacy of the proposed framework by using three benchmark reliability analysis problems from the literature. The results obtained indicate the accuracy and efficiency of the proposed approach compared to already established reliability analysis methods in the literature.