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

TL;DR

This paper introduces F2NARX, a novel probabilistic function-on-function nonlinear autoregressive model that significantly improves efficiency and accuracy in emulating dynamical systems, enabling effective reliability analysis with limited data.

Contribution

The paper proposes F2NARX, a new function-on-function NARX model that combines PCA and Gaussian processes for fast, accurate, and probabilistic dynamical system emulation.

Findings

F2NARX outperforms existing NARX models in efficiency by orders of magnitude.

F2NARX achieves higher accuracy in response prediction.

Probabilistic predictions enable reliable failure probability estimation with limited data.

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.