Gaussian-process surrogate indicators for residual-based adaptive GMsFEM

Residual-based adaptive GMsFEM for high-contrast elliptic problems repeatedly evaluates local weighted H-1 indicators on every coarse neighborhood, making indicator evaluation a recurring cost in repeated-query settings. We introduce a non-intrusive Gaussian-process (GP) surrogate for the indicator scores used in D"orfler marking. The operational predictor is the GP posterior mean, algebraically equivalent to a kernel ridge regression (KRR) estimator under the stated convention; it uses compressed local solution and spectral features without changing the multiscale solve, local spectral construction, or basis enrichment. A nonuniform perturbed-marking result quantifies how pointwise score errors affect the exact indicator mass captured by surrogate-selected neighborhoods, while a conditional bounded-discrepancy KRR pathway identifies sufficient assumptions for such score bounds. In controlled held-out in-distribution tests, the surrogate-guided method gives error-versus-DoF trends comparable with classical H-1-residual offline adaptivity and evaluates the online indicator component 2.0-2.1 times faster, excluding offline data generation and GP training.