Koopman operator for time-dependent reliability analysis

TL;DR

This work tackles time-dependent reliability analysis for nonlinear dynamical systems by employing a data-driven Koopman framework that learns observables enabling linear evolution in a lifted space. An end-to-end DL architecture encodes the state into Koopman coordinates, applies a linear operator $\mathcal{K}$, and decodes back to predict future states, with a loss $\mathcal{L}=\lambda_1\mathcal{L}_1+\lambda_2\mathcal{L}_2+\lambda_3\mathcal{L}_3$ to ensure reconstruction, linearity, and accurate time-prediction. Two architectures address uncertainty in initial conditions and in system parameters, demonstrating robust first passage failure time PDFs across ODE and PDE examples, including chaotic systems like the Lorenz attractor. Compared to pure data-driven models, the Koopman approach yields higher accuracy, better generalization, and successful out-of-distribution predictions, highlighting its practical potential for reliability assessment under uncertainty in engineering systems. The results indicate the method’s scalability to PDEs and its applicability to time-dependent reliability problems where first passage analysis is essential.

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.

Figures (21)

• Figure 1: Schematic representation of the proposed architecture for systems with uncertainty in the initial conditions, consists of encoder, decoder and Koopman operator. Training of encoder, Koopman operator and decoder are done simultaneously.
• Figure 2: Schematic representation of the proposed architecture for systems with uncertainty in the parameters, consists of encoder, decoder and Koopman operator. State variables appended to system parameters is fed as input to the encoder. Input to the decoder is concatenated vector of Koopman coordinates of successive time step and system parameters
• Figure 3: Architecture of proposed Koopman framework (a fully connected network) for duffing oscillator example with uncertainty in the initial conditions
• Figure 4: State variable of duffing oscillator predicted for random initial conditions; (a) variation of displacement ($x$) with time, (b) variation of velocity ($\dot x$) with time
• Figure 5: Convergence of prediction error ($\epsilon$) with number of training samples $N_s$ for the case of duffing oscillator with random initial conditions