# Gaussian-process surrogate indicators for residual-based adaptive GMsFEM

arXiv: https://arxiv.org/abs/2609.26843 · HTML: https://mlsanity.com/paper/2609.26843/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 33 references that are in the map.
- First posted on arXiv 22 Sep 2026.

## Builds on (most cited first, 7 of 7)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Deterministic error bounds for kernel-based learning techniques under bounded noise](https://mlsanity.com/paper/2008.04005/) (2020)
- [Generalized Multiscale Finite Element Methods (GMsFEM)](https://mlsanity.com/paper/1301.2866/) (2013)
- [Operator Learning with Gaussian Processes](https://mlsanity.com/paper/2409.04538/) (2024)
- [AMS-Net: Adaptive Multiscale Sparse Neural Network with Interpretable Basis Expansion for Multiphase Flow Problems](https://mlsanity.com/paper/2207.11735/) (2022)

## Abstract (arXiv)

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.