Evaluation of Gaussian process regression kernel functions for improving groundwater prediction

Systematic model error is caused by the unreasonable simplification of real groundwater system, which damages the reliability of groundwater model prediction. Gaussian process regression (GPR) is a popular data-driven method used to build a statistical complementary model to correct systematic prediction error and improve model prediction. Kernel function is a crucial component of GPR, it represents the assumptions on systematic prediction error that we want to learn. In order to evaluate the influence of GPR kernel on capturing systematic prediction error, we compared the performances of GPR with nine kernels in groundwater predictions. These kernels include three commonly used single kernels, i.e., squared exponential (SE), Matern class (MT), and rational quadratic (RQ) kernels, and six new kernels formed by combining these single kernels as sum or product of pairs (SE_MT_s, SE_RQ_s, MT_RQ_s, SE_MT_p, SE_RQ_p, and MT_RQ_p). Moreover, three groundwater case studies, including two analytical cases and one real case, are used to test these kernels, and four metrics are applied to measure the performances of these kernels. The evaluation results show that kernel functions have apparent influence on the ability of the GPR-based complementary model to capture the systematic prediction error of groundwater model. The single kernel RQ-based GPR yields the best performance in improving groundwater predictions among the nine kernels for all case studies. The combined kernels MT_RQ_s, SE_RQ_s also perform well in groundwater predictions, and the commonly used SE kernel has mediocre performance in predicting groundwater observations.