GAR: Generalized Autoregression for Multi-Fidelity Fusion
Authors: Yuxin Wang, Zheng Xing, WEI XING

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 same predictive mean accuracy but requires signiﬁcantly less computation. In experiments of canonical PDEs and scientiﬁc computational examples, the proposed method consistently outperforms the SOTA methods with a large margin (up to 6x improvement in RMSE) with only a few high-ﬁdelity training samples.