Probabilistic function-on-function nonlinear autoregressive model for emulation and reliability analysis of dynamical systems

TL;DR

The paper tackles the challenge of efficiently emulating strongly nonlinear dynamical systems with uncertainty. It develops F2NARX, a probabilistic function-on-function nonlinear autoregressive surrogate that uses a one-time-window-ahead strategy, PCA-based local feature extraction, and sparse Gaussian process regression to enable fast, uncertainty-aware predictions. A probabilistic prediction scheme based on the unscented transform is introduced, enabling active learning for reliable first-passage probability estimation. Demonstrations on Bouc–Wen and a nonlinear steel frame show substantial speedups and accuracy improvements over existing NARX variants, with active learning significantly reducing necessary training data for reliability analysis, highlighting practical impact for design, control, and digital-twin applications.

Abstract

Constructing accurate and computationally efficient surrogate models (or emulators) for predicting dynamical system responses is critical in many engineering domains, yet remains challenging due to the strongly nonlinear and high-dimensional mapping from external excitations and system parameters to system responses. This work introduces a novel Function-on-Function Nonlinear AutoRegressive model with eXogenous inputs (F2NARX), which reformulates the conventional NARX model from a function-on-function regression perspective, inspired by the recently proposed $\mathcal{F}$-NARX method. The proposed framework substantially improves predictive efficiency while maintaining high accuracy. By combining principal component analysis with Gaussian process regression, F2NARX further enables probabilistic predictions of dynamical responses via the unscented transform in an autoregressive manner. The effectiveness of the method is demonstrated through case studies of varying complexity. Results show that F2NARX outperforms state-of-the-art NARX model by orders of magnitude in efficiency while achieving higher accuracy in general. Moreover, its probabilistic prediction capabilities facilitate active learning, enabling accurate estimation of first-passage failure probabilities of dynamical systems using only a small number of training time histories.

Figures (14)

• Figure 1: The illustration of F2NARX model.
• Figure 2: Illustration of probabilistic prediction approach of the F2NARX model.
• Figure 3: Flowchart of active learning-based dynamic reliability analysis.
• Figure 4: Five different realizations of excitations $u(t)$ and corresponding responses $y(t)$ for the Bouc-Wen oscillator.
• Figure 5: Plots of prediction error and number of principal components of local response function $y(t)$ as functions of the time window width $T$ and retained variance proportion $\varepsilon_{\lambda}$.