GAR：用于多保真度融合的广义自回归
GAR: Generalized Autoregression for Multi-Fidelity Fusion
摘要：
在许多科学研究和工程应用中，需要对复杂系统进行反复仿真，通常采用代理来快速估计整个系统。为了降低生成训练示例的昂贵成本，将低保真度（快速但不准确）和高保真度（缓慢但准确）模拟的结果结合起来已经成为一种很有前途的方法。尽管多保真度融合技术发展迅速，但大多数现有方法都需要特定的数据结构，并且不能很好地扩展到高维输出。为了解决这些问题，我们推广了经典的自回归（AR），这是广泛使用的，由于其简单性，鲁棒性，准确性和易处理性，并提出广义自回归（GAR）使用张量公式和潜在的功能。GAR可以处理任意维的输出和任意的多保真度数据结构，满足复杂问题的多保真度融合需求;它具有完全易处理的似然性和后验，不需要近似推理，并且可以很好地扩展到高维问题。此外，我们证明了autokrigeability定理的基础上GAR在多保真度的情况下，并开发CIGAR，一个简化的GAR与精确的预测平均精度与计算减少了一个因素的d 3，其中d是输出的维数。实证评估包括许多典型的偏微分方程和真实的科学例子，并表明所提出的方法始终优于SOTA方法，具有很大的利润率（RMSE提高高达6倍），只有一对夫妇的高保真训练样本。
英文摘要：
In many scientific research and engineering applications where repeated simulations of complex systems are conducted, a surrogate is commonly adopted to quickly estimate the whole system. To reduce the expensive cost of generating training examples, it has become a promising approach to combine the results of low-fidelity (fast but inaccurate) and high-fidelity (slow but accurate) simulations. Despite the fast developments of multi-fidelity fusion techniques, most existing methods require particular data structures and do not scale well to high-dimensional output. To resolve these issues, we generalize the classic autoregression (AR), which is wildly used due to its simplicity, robustness, accuracy, and tractability, and propose generalized autoregression (GAR) using tensor formulation and latent features. GAR can deal with arbitrary dimensional outputs and arbitrary multifidelity data structure to satisfy the demand of multi-fidelity fusion for complex problems; it admits a fully tractable likelihood and posterior requiring no approximate inference and scales well to high-dimensional problems. Furthermore, we prove the autokrigeability theorem based on GAR in the multi-fidelity case and develop CIGAR, a simplified GAR with the exact predictive mean accuracy with computation reduction by a factor of d 3, where d is the dimensionality of the output. The empirical assessment includes many canonical PDEs and real scientific examples and demonstrates that the proposed method consistently outperforms the SOTA methods with a large margin (up to 6x improvement in RMSE) with only a couple high-fidelity training samples.
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