## 3 Key Processes for Deep Learning-Based Seepage Field Prediction
### Comparative Framework for Deep Learning Architectures in Seepage Modeling  
1. Analysis of Differences in Mathematical Structures  
1. FeedForward Neural (FFN) networks Mathematical expression: \[y=\sigma(W_{n}\cdot\sigma(W_{n-1}\cdots\sigma(W_{1}x+b_{1}))+b_{n})\] (1) Explanation: purely data-driven layer-by-layer nonlinear transformations with no spatial/temporal structure awareness. Structural Features: FFN networks pass information through multiple fully connected layers (Dense Layer) step by step, and each layer applies a nonlinear activation function (e.g., ReLU, Sigmoid) to the inputs after a linear transformation (weight matrix W with bias b), which are essentially higher-order nonlinear function fitters capable of capturing complex mapping relationships between input features and target variables. Limitations in seepage field applications: (i) Blind-box modeling: FFN networks rely on end-to-end data-driven modeling and lack explicit encoding of spatial topology or physical laws, leading to unphysical results (e.g., negative pressure regions) when predicting continuous fields (e.g., pressure distributions); (ii) Scenario adaptation: suitable for single-point parameter prediction (e.g., WC in a single well), but low reconstruction resolution due to neglect of local spatial correlations when dealing with gridded field data [10]. * 2. Convolutional Neural Networks, CNNs Mathematical expression: \[F_{\text{out}}(i,j)=\sum_{m}\sum_{k,l}W_{m}(k,l)\cdot F_{\text{in}}(i+k,j+l)+b\] (2) Explanation: extraction of local spatial features by convolution kernel for gridded data (e.g., permeability field images). Structural features: CNNs extract local spatial features by sliding convolutional kernels (e.g., \(3\times 3\) filters) and realize feature dimensionality reduction and translation invariance by using a Pooling operation (Pooling). A CNN's local sensing mechanism is naturally adapted to gridded data (e.g., permeability field images) and is capable of capturing spatial patterns such as inter-well disturbances and hyper-permeable channels. Advantages in seepage field applications: (i) \"Pictorial\" modeling: Treating the seepage field as a 2D/3D image, spatial heterogeneity can be automatically identified by convolution kernels (e.g., saturation field reconstruction in [PERSON] et al. [3]); (ii) Potential for physical customization: By designing specific convolutional kernels (e.g., difference kernels), physical laws (e.g., Darcy's law gradient computation) can be partially embedded to improve the consistency of predictions (e.g., PD-CNN in [PERSON] et al. [2]). * 3. Graph Neural Networks, GNNs Mathematical expression: \[h_{v}^{(k)}=\phi\left(h_{v}^{(k-1)},\bigoplus_{u\in\mathcal{N}(v)}\psi\left(h_ {u}^{(k-1)}\right)\right)\] (3) Explanation: processing of non-Euclidean spatial data (e.g., fractured networks) by node aggregation (Aggregation) and message passing (Message Passing). Structural features: GNNs abstract the seepage system as a graph structure (nodes = wells/pores, edges = flow paths) and dynamically update the node states through Message Passing and node aggregation (Aggregation). Their non-Euclidean data-processing capability is particularly suitable for describing complex topologies such as fracture networks and unstructured grids. Uniqueness in seepage field applications: (i) \"Social network\" modeling: Similar to the propagation of user relationships in social networks, GNNs simulate fluid flow paths along a fracture or pore network (e.g., the microscopic seepage simulation of [PERSON] et al. [16]); (ii) Multi-scale fusion: The macroscopic well network layout and microscopic pore structure can be encoded simultaneously to realize cross-scale seepage coupling ([PERSON] et al. [18] for fracture-matrix interaction modeling). * 4.