Data-Free and Data-Efficient Physics-Informed Neural Network Approaches to Solve the Buckley–Leverett Problem

Physics-informed neural networks (PINNs) are an emerging technology in the scientific computing domain. Contrary to data-driven methods, PINNs have been shown be able approximate and generalize well a wide range of partial differential equations (PDEs) by imbedding underlying physical laws describing PDE. PINNs, however, can struggle with modeling hyperbolic conservation that develop shocks, classic example this is Buckley–Leverett problem for fluid flow porous media. In work, we explore specialized network architectures shock front. We present extensions standard multilayer perceptron (MLP) inspired attention mechanism. The attention-based model was, compared model, results show architecture more robust solving problem, data-efficient, accurate. Moreover, utilizing distance functions, obtain truly data-free solutions problem. approach, initial boundary conditions (I/BCs) imposed hard manner as opposed soft manner, where labeled data provided on I/BCs. This allows us use substantially smaller NN solution