When repeated evaluations for varying parameter configurations of a high-fidelity physical model are required, surrogate modeling techniques based on model order reduction are desired. In absence of the governing equations describing the dynamics, we need to construct the parametric reduced-order surrogate model in a non-intrusive fashion. In this setting, the usual residual-based error estimate for optimal parameter sampling associated with the reduced basis method is not directly available. Our work provides a non-intrusive optimality criterion to efficiently populate the parameter snapshots, thereby, enabling us to effectively construct a parametric surrogate model. We consider separate parameter-specific proper orthogonal decomposition (POD) subspaces and propose an active-learning-driven surrogate model using kernel-based shallow neural networks, abbreviated as ActLearn-POD-KSNN surrogate model. To demonstrate the validity of our proposed ideas, we present numerical experiments using two physical models, namely Burgers' equation and shallow water equations. Both the models have mixed -- convective and diffusive -- effects within their respective parameter domains, with each of them dominating in certain regions. The proposed ActLearn-POD-KSNN surrogate model efficiently predicts the solution at new parameter locations, even for a setting with multiple interacting shock profiles.

翻译：当需要对高保真物理模型在不同参数配置下进行重复评估时，基于模型降阶的代理建模技术具有重要应用价值。在缺乏描述动力学过程的控制方程的情况下，需要以非侵入式方式构建参数化降阶代理模型。在此设定中，传统基于残差的误差估计方法（用于简化基方法中的最优参数采样）无法直接使用。本研究提出了一种非侵入式最优性准则，用于高效填充参数快照，从而有效构建参数化代理模型。我们考虑各参数特有的本征正交分解（POD）子空间，并提出了基于核的浅层神经网络的主动学习驱动代理模型，简称ActLearn-POD-KSNN代理模型。为验证所提方法的有效性，我们采用两个物理模型（即Burgers方程和浅水方程）进行数值实验。这两个模型在其参数域内均具有对流与扩散混合效应，且不同区域的主导效应各异。实验表明，所提出的ActLearn-POD-KSNN代理模型能够高效预测新参数位置的解，即使面对包含多个相互作用激波剖面的复杂场景仍能保持良好的预测性能。