Prediction of Porous Media Fluid Flow with Spatial Heterogeneity Using Criss-Cross Physics-Informed Convolutional Neural Networks

Recent advances in deep neural networks have shed new light on physics, engineering, and scientific computing. Reconciling the data-centered viewpoint with physical simulation is one of the research hotspots. The physicsinformed neural network (PINN) is currently the most general framework, which is more popular due to the convenience of constructing NNs and excellent generalization ability. The automatic differentiation (AD)-based PINN model is suitable for the homogeneous scientific problem; however, it is unclear how AD can enforce flux continuity across boundaries between cells of different properties where spatial heterogeneity is represented by grid cells with different physical properties. In this work, we propose a criss-cross physics-informed convolutional neural network (CC-PINN) learning architecture, aiming to learn the solution of parametric PDEs with spatial heterogeneity of physical properties. To achieve the seamless enforcement of flux continuity and integration of physical meaning into CNN, a predefined 2D convolutional layer is proposed to accurately express transmissibility between adjacent cells. The efficacy of the proposed method was evaluated through predictions of several petroleum reservoir problems with spatial heterogeneity and compared against state-of-the-art (PINN) through numerical analysis as a benchmark, which demonstrated the superiority of the proposed method over the PINN.