LIMITATIONS OF PHYSICS INFORMED MACHINE LEARNING FOR NONLINEAR TWO-PHASE TRANSPORT IN POROUS MEDIA

Abstract

Deep learning techniques have recently been applied to a wide range of computational physics problems. In this paper, we focus on developing a physics-based approach that enables the neural network to learn the solution of a dynamic fluid-flow problem governed by a nonlinear partial differential equation (PDE). The main idea of physics informed machine learning (PIML) approaches is to encode the underlying physical law (i.e., the PDE) into the neural network as prior information. We investigate the applicability of the PIML approach to the forward problem of immiscible two-phase fluid transport in porous media, which is governed by a nonlinear first-order hyperbolic PDE subject to initial and boundary data. We employ the PIML strategy to solve this forward problem without any additional labeled data in the interior of the domain. Particularly, we are interested in nonconvex flux functions in the PDE, where the solution involves shocks and mixed waves (shocks and rarefactions). We have found that such a PIML approach fails to provide reasonable approximations to the solution in the presence of shocks in the saturation field. We investigated several architectures and experimented with a large number of neural-network parameters, and the overall finding is that PIML strategies that employ the nonlinear hyperbolic conservation equation in the loss function are inadequate. However, we have found that employing a parabolic form of the conservation equation, whereby a small amount of diffusion is added, the neural network is consistently able to learn accurate approximation of the solutions containing shocks and mixed waves.

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