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.

翻译：构建精确且计算高效的代理模型（或仿真器）以预测动力学系统响应，在许多工程领域中至关重要，但由于从外部激励和系统参数到系统响应的映射具有强非线性和高维特性，这仍然是一项挑战。本研究受近期提出的$\mathcal{F}$-NARX方法启发，从函数对函数回归的视角重构了传统的NARX模型，提出了一种新颖的带外生输入的函数对函数非线性自回归模型（F2NARX）。该框架在保持高精度的同时，显著提升了预测效率。通过将主成分分析与高斯过程回归相结合，F2NARX进一步能够通过无迹变换以自回归方式实现动力学响应的概率预测。该方法在不同复杂度的案例研究中得到了验证。结果表明，F2NARX在效率上比最先进的NARX模型高出数个数量级，同时通常能达到更高的精度。此外，其概率预测能力促进了主动学习，使得仅使用少量训练时间历程即可准确估计动力学系统的首次穿越失效概率。