Improved physics-informed neural networks for the reinterpreted discrete fracture model

This paper is the first attempt to apply improved-physics-informed neural networks (I-PINNs) to simulate fluid flow in fractured porous media based on the reinterpreted discrete fracture model (RDFM). The RDFM, first introduced by Xu and Yang, is a hybrid-dimensional model where Dirac-delta functions are used to characterize fractures and superposed with the permeability tensor. In this paper, we apply the physical information neural networks (PINNs) to RDFM. Different from the traditional PINNs where the PDE residual was used as the loss function, we adopt the finite element discretization of RDFM to build the loss function, avoiding the large gradient problem and difficulties in automatic differentiation. This new method is named as the improved PINNs (I-PINNs). Moreover, we combine the RDFM with incompressible miscible displacement in porous media. The bound-preserving technique of the I-PINNs is proposed and applied to the coupled system mentioned above, keeping the numerical concentration to be between 0 and 1. It is worth noting that one of the advantages of I-PINNs compared to PINNs is that it can better capture the pressure gradient at the fractures. Compared with traditional finite element methods for flow equations, I-PINNs do not request the inversion of the stiffness matrix. In addition, different from the traditional bound-preserving technique for contaminant transportation, I-PINNs preserve the physical bounds without taking a limited time step. Several numerical experiments are given to verify the feasibility and accuracy of the I-PINNs.