Deep ensemble learning for high-dimensional subsurface fluid flow modeling

The accuracy of Deep Learning (DL) algorithms can be improved by combining several deep learners into an ensemble. This avoids the continuous endeavor required to adjust the architecture of individual networks or the nature of the propagation. This study investigates prediction improvements possible using Deep Ensemble Learning (DEL) to determine four distinct multiscale basis functions in the mixed Generalized Multiscale Finite Element Method (GMsFEM), involving the permeability field as the only input. 376,250 samples were initially generated, filtered down to 367,811 after data pre-processing. A standard Convolutional Neural Network (CNN) named SkiplessCNN and three skip connection-based CNNs named FirstSkipCNN, MidSkipCNN, and DualSkipCNN were developed for the base learners. For each basis function, these four CNNs were combined into an ensemble model using linear regression and ridge regression, separately, as part of the stacking technique. A comparison of the coefficient of determination (R²) and Mean Squared Error (MSE) confirms the effectiveness of all three skip connections in enhancing the performance of the standard CNN, with DualSkip being the most effective among them. Additionally, as evaluated on the testing subset, the combined models meaningfully outperform the individual models for all basis functions. The case that applies linear regression delivers R² ranging from 0.8456 to 0.9191 and MSE ranging from 0.0092 to 0.0369. The ridge regression case achieves marginally better predictions with R² ranging from 0.8539 to 0.922, and MSE ranging from 0.009 to 0.0349 because its solution involves more evenly distributed weights.