自适应多保真度高斯过程在噪声训练数据下的性能评估：统计分析
Assessing the Performance of an Adaptive Multi-Fidelity Gaussian Process with Noisy Training Data: A Statistical Analysis
摘要：
尽管增加了计算资源，但从计算的角度来看，基于仿真的设计优化（SBDO）过程可能非常昂贵，特别是如果需要高保真求解器。多保真度元模型已成功地应用于减少SBDO过程的计算成本。在这种情况下，本文提出了一个自适应多保真度元模型的基础上高斯过程回归（MF-GPR）的噪声数据的性能评估。MF-GPR被开发为：（i）管理任意数量的保真度水平，（ii）处理受噪声影响的目标函数评估，以及（iii）通过自适应采样来提高其拟合精度。多保真度是通过将低保真度元模型与连续保真度水平之间的误差的元模型桥接来实现的。MF-GPR通过回归处理数值噪声。自适应采样方法基于最大预测不确定性，并包括自动选择采样保真度的规则。MF-GPR的性能进行了评估的一组五个分析基准问题的影响，嘈杂的目标函数评价。由于噪声在目标函数的评估中引入了随机性，因此采用统计分析方法来评估MF-GPR的性能和鲁棒性。本文讨论了MF-GPR在全局逼近目标函数和识别全局极小值方面的效率和有效性。使用一个、两个和三个保真度水平。统计分析的结果表明，与使用一个或两个保真度相比，使用三个保真度水平实现了无噪声目标函数的更准确的全局表示。
英文摘要：
Despite the increased computational resources, the simulation-based design optimization (SBDO) procedure can be very expensive from a computational viewpoint, especially if high-fidelity solvers are required. Multi-fidelity metamodels have been successfully applied to reduce the computational cost of the SBDO process. In this context, the paper presents the performance assessment of an adaptive multi-fidelity metamodel based on a Gaussian process regression (MF-GPR) for noisy data. The MF-GPR is developed to: (i) manage an arbitrary number of fidelity levels, (ii) deal with objective function evaluations affected by noise, and (iii) improve its fitting accuracy by adaptive sampling. Multi-fidelity is achieved by bridging a low-fidelity metamodel with metamodels of the error between successive fidelity levels. The MF-GPR handles the numerical noise through regression. The adaptive sampling method is based on the maximum prediction uncertainty and includes rules to automatically select the fidelity to sample. The MF-GPR performance are assessed on a set of five analytical benchmark problems affected by noisy objective function evaluations. Since the noise introduces randomness in the evaluation of the objective function, a statistical analysis approach is adopted to assess the performance and the robustness of the MF-GPR. The paper discusses the efficiency and effectiveness of the MF-GPR in globally approximating the objective function and identifying the global minimum. One, two, and three fidelity levels are used. The results of the statistical analysis show that the use of three fidelity levels achieves a more accurate global representation of the noise-free objective function compared to the use of one or two fidelities.
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