In this study, we propose a novel surrogate modelling approach to efficiently and accurately approximate the response of complex dynamical systems driven by time-varying exogenous excitations over extended time periods. Our approach, that we name \emph{manifold nonlinear autoregressive modelling with exogenous input} (mNARX), involves constructing a problem-specific exogenous input manifold that is optimal for constructing autoregressive surrogates. The manifold, which forms the core of mNARX, is constructed incrementally by incorporating the physics of the system, as well as prior expert- and domain- knowledge. Because mNARX decomposes the full problem into a series of smaller sub-problems, each with a lower complexity than the original, it scales well with the complexity of the problem, both in terms of training and evaluation costs of the final surrogate. Furthermore, mNARX synergizes well with traditional dimensionality reduction techniques, making it highly suitable for modelling dynamical systems with high-dimensional exogenous inputs, a class of problems that is typically challenging to solve.Since domain knowledge is particularly abundant in physical systems, such as those found in engineering applications, mNARX is well suited for these applications. We demonstrate that mNARX outperforms traditional autoregressive surrogates in predicting the response of a classical coupled spring-mass system excited by a one-dimensional random excitation. Additionally, we show that mNARX is well suited for emulating very high-dimensional time- and state-dependent systems, even when affected by active controllers, by surrogating the dynamics of a realistic aero-servo-elastic onshore wind turbine simulator. In general, our results demonstrate that mNARX offers promising prospects for modelling complex dynamical systems, in terms of accuracy and efficiency.

翻译：在本研究中，我们提出了一种新颖的代理建模方法，该方法能够高效且准确地近似由时变外生激励驱动的复杂动力系统在长时间跨度内的响应。我们将这种方法命名为"流形非线性自回归外生输入模型"（manifold nonlinear autoregressive modelling with exogenous input, mNARX），其核心在于构建一个特定于问题的最优外生输入流形，以用于构建自回归代理模型。该流形作为mNARX的核心，通过逐步融入系统物理特性以及先验专家知识与领域知识来构建。由于mNARX将完整问题分解为一系列复杂度低于原问题的子问题，因此它在最终代理模型的训练与评估成本方面均能随问题复杂度良好扩展。此外，mNARX与传统降维技术具有协同效应，这一特性使其尤其适用于对高维外生输入的动力系统进行建模——这类问题通常具有求解挑战性。鉴于领域知识在物理系统中尤为丰富（例如工程应用中的系统），mNARX非常适合此类应用。我们通过经典耦合弹簧-质量系统（受一维随机激励）的响应预测实验，证明了mNARX优于传统自回归代理模型。同时，通过代理一台真实气动-伺服-弹性陆上风力发电机模拟器的动力学行为，我们展示了mNARX在模拟受主动控制器影响的高维时变与状态依赖系统方面的优异能力。总体而言，我们的结果表明mNARX在精度与效率方面为复杂动力系统建模提供了极具前景的方案。