Simulating complex physical processes across a domain of input parameters can be very computationally expensive. Multi-fidelity surrogate modeling can resolve this issue by integrating cheaper simulations with the expensive ones in order to obtain better predictions at a reasonable cost. We are specifically interested in computer experiments where real-valued fidelity parameters determine the fidelity of the numerical output, such as finite element methods. In these cases, integrating this fidelity parameter in the analysis enables us to make inference on fidelity levels that have not been observed yet. Such models have been developed, and we propose a new adaptive non-stationary kernel function which more accurately reflects the behavior of computer simulation outputs. In addition, we develop an active learning strategy based on the integrated mean squared prediction error (IMSPE) to identify the best design points across input parameters and fidelity parameters, while taking into account the computational cost associated with the fidelity parameters. We illustrate this methodology through numerical examples and applications to finite element methods. An $\textsf{R}$ package for the proposed methodology is provided in an open repository.

翻译：在输入参数域内模拟复杂物理过程可能计算成本极高。多保真度代理建模通过整合廉价模拟与昂贵模拟，能够以合理成本获得更优预测，从而解决这一问题。我们特别关注计算机实验，其中实值保真度参数（如有限元方法中的参数）决定数值输出的保真度。在此类场景中，将保真度参数纳入分析使我们能够对尚未观测到的保真度水平进行推断。现有模型已有所发展，我们提出一种新的自适应非平稳核函数，能更准确地反映计算机模拟输出的行为特征。此外，我们基于集成均方预测误差（IMSPE）开发了一种主动学习策略，该策略在考虑保真度参数相关计算成本的同时，能够识别输入参数和保真度参数范围内的最优设计点。我们通过数值算例及有限元方法应用展示了该方法。相关开源仓库提供了实现该方法的$\textsf{R}$软件包。