# Koopman operator for time-dependent reliability analysis

arXiv: https://arxiv.org/abs/2203.02658 · HTML: https://mlsanity.com/paper/2203.02658/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 52 references that are in the map.
- First posted on arXiv 5 Mar 2022.

## Builds on (most cited first, 8 of 8)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [AR-Net: A simple Auto-Regressive Neural Network for time-series](https://mlsanity.com/paper/1911.12436/) (2019)
- [Assessment of DeepONet for reliability analysis of stochastic nonlinear dynamical systems](https://mlsanity.com/paper/2201.13145/) (2022)

## Built on by (most cited first, 7 of 7)

- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [A foundational neural operator that continuously learns without forgetting](https://mlsanity.com/paper/2310.18885/) (2023)
- [Time Marching Neural Operator FE Coupling: AI Accelerated Physics Modeling](https://mlsanity.com/paper/2504.11383/) (2025)
- [Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured grid](https://mlsanity.com/paper/2409.00604/) (2024)
- [High Energy Density Radiative Transfer in the Diffusion Regime with Fourier Neural Operators](https://mlsanity.com/paper/2405.04003/) (2024)
- [Harnessing physics-informed operators for high-dimensional reliability analysis problems](https://mlsanity.com/paper/2409.04708/) (2024)
- [CoNBONet: Conformalized Neuroscience-inspired Bayesian Operator Network for Reliability Analysis](https://mlsanity.com/paper/2603.21678/) (2026)

## Abstract (arXiv)

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.