A Quantitative Insight Into the Role of Skip Connections in Deep Neural Networks of Low Complexity: A Case Study Directed at Fluid Flow Modeling

Deep feed-forward networks, with high complexity, backpropagate the gradient of the loss function from final layers to earlier layers. As a consequence, the “gradient” may descend rapidly toward zero. This is known as the vanishing gradient phenomenon that prevents earlier layers from benefiting from further training. One of the most efficient techniques to solve this problem is using skip connection (shortcut) schemes that enable the gradient to be directly back-propagated to earlier layers. This paper investigates whether skip connections significantly affect the performance of deep neural networks of low complexity or whether their inclusion has little or no effect. The analysis was conducted using four Convolutional Neural Networks (CNNs) to predict four different multiscale basis functions for the mixed Generalized Multiscale Finite Element Method (GMsFEM). These models were applied to 249,375 samples. Three skip connection schemes were added to the base structure: Scheme 1 from the first convolutional block to the last, Scheme 2 from the middle to the last block, and Scheme 3 from the middle to the last and the second-to-last blocks. The results demonstrate that the third scheme is most effective, as it increases the coefficient of determination (R²) value by 0.0224–0.044 and decreases the Mean Squared Error (MSE) value by 0.0027–0.0058 compared to the base structure. Hence, it is concluded that enriching the last convolutional blocks with the information hidden in neighboring blocks is more effective than enriching using earlier convolutional blocks near the input layer.