A layer-specific constraint-based enriched physics-informed neural network for solving two-phase flow problems in heterogeneous porous media

In this study, we propose a constraint learning strategy based on interpretability analysis to improve the convergence and accuracy of the enriched physics-informed neural network (EPINN), which is applied to simulate two-phase flow in heterogeneous porous media. Specifically, we first analyze the layerwise outputs of EPINN, and identify the distinct functions across layers, including dimensionality adjustment, pointwise construction of non-equilibrium potential, extraction of high-level features, and the establishment of long-range dependencies. Then, inspired by these distinct modules, we propose a novel constraint learning strategy based on regularization approaches, which improves neural network (NN) learning through layer-specific differentiated updates to enhance cross-timestep generalization. Since different neural network layers exhibit varying sensitivities to global generalization and local regression, we decrease the update frequency of layers more sensitive to local learning under this constraint learning strategy. In other words, the entire neural network is encouraged to extract more generalized features. The superior performance of the proposed learning strategy is validated through evaluations on numerical examples with varying computational complexities. Post hoc analysis reveals that gradient propagation exhibits more pronounced staged characteristics, and the partial differential equation (PDE) residuals are more uniformly distributed under the constraint guidance. Interpretability analysis of the adaptive constraint process suggests that maintaining a stable information compression mode facilitates progressive convergence acceleration.