Data-driven surrogate models are widely used for applications such as design optimization and uncertainty quantification, where repeated evaluations of an expensive simulator are required. For most partial differential equation (PDE) simulations, the outputs of interest are often spatial or spatial-temporal fields, leading to very high-dimensional outputs. Despite the success of existing data-driven surrogates for high-dimensional outputs, most methods require a significant number of samples to cover the response surface in order to achieve a reasonable degree of accuracy. This demand makes the idea of surrogate models less attractive considering the high-computational cost to generate the data. To address this issue, we exploit the multifidelity nature of a PDE simulation and introduce deep coregionalization, a Bayesian nonparametric autoregressive framework for efficient emulation of spatial-temporal fields. To effectively extract the output correlations in the context of multifidelity data, we develop a novel dimension reduction technique, residual principal component analysis. Our model can simultaneously capture the rich output correlations and the fidelity correlations and make high-fidelity predictions with only a small number of expensive, high-fidelity simulation samples. We show the advantages of our model in three canonical PDE models and a fluid dynamics problem. The results show that the proposed method can not only approximate simulation results with significantly less cost (by bout 10%-25%) but also further improve model accuracy.

数据驱动的替代模型广泛用于诸如设计优化和不确定性量化之类的应用中，在这些应用中，需要对昂贵的模拟器进行反复评估。对于大多数偏微分方程（PDE）模拟，关注的输出通常是空间或时空场，从而导致非常高维的输出。尽管现有的数据驱动的替代品已成功用于高维输出，但大多数方法仍需要大量样本才能覆盖响应表面，以实现合理的准确性。考虑到生成数据的高计算成本，这种需求使得代理模型的想法不那么有吸引力。为了解决这个问题，我们利用了PDE模拟的多保真性，并介绍了深度共区域化，一种有效模拟时空场的贝叶斯非参数自回归框架。为了在多保真度数据的背景下有效地提取输出相关性，我们开发了一种新的降维技术，即残留主成分分析。我们的模型可以同时捕获丰富的输出相关性和保真度相关性，并且仅使用少量昂贵的高保真度仿真样本即可进行高保真度预测。我们在三个规范的PDE模型和流体动力学问题中展示了我们模型的优势。结果表明，所提出的方法不仅可以以低得多的成本逼近模拟结果（约10％-25％），而且可以进一步提高模型的准确性。