Variational Interface Physics Informed Neural Networks (VI-PINNs) for Heterogeneous Subsurface Systems

ABSTRACT This study presents a novel approach to Physics Informed Neural Networks (PINNs) called Variational Interface PINNs (VI-PINNs) for modeling physical systems with material interfaces. In conventional PINNs, a loss functional obtained from the residual of the strong form of governing equations is minimized to estimate the variable of interest. By contrast, the presented approach relies on minimizing the potential energy functional of a physical system using artificial neural networks. Firstly, we present appropriate modifications to the neural network architecture so that material interfaces can be captured effectively in a variational framework using neural networks. Secondly, we incorporate the interface coupling conditions in the energy functionals to be minimized using VI-PINNs. Thirdly, we examine several numerical integration techniques and compare their accuracy and cost to evaluate the Lagrangians in the VI-PINNs framework. Finally, we discuss a variationally consistent penalty approach to enforce boundary and interface conditions better. Several benchmark 1D and 2D examples are solved using the proposed method and conventional PINNs. Their performances are then compared in terms of cost and accuracy, and their relative strengths and weaknesses are identified. INTRODUCTION Subsurface systems are often characterized by a high degree of heterogeneity. Heterogeneities in the subsurface may arise from the natural layering of the earth's crust or naturally occurring faults. In addition, stimulation methods such as hydraulic fracturing also contribute to its heterogeneous character. The heterogeneity of these systems can significantly impact the behavior and movement of fluids, such as water, oil, and gas, within the subsurface. As a result, the study and understanding of heterogeneous subsurface systems are critical for a wide range of industries, including petroleum engineering, geotechnical engineering, hydrology, and environmental science, where accurate predictions of fluid behavior and movement are essential. Traditionally, several high-fidelity physics-based numerical modeling techniques, including the Finite Element Method (FEM), have been utilized for modeling subsurface systems. These methods involve using a mesh or grid to approximate the system and necessitate the proper meshing of the subsurface into non-overlapping elements. However, generating meshes that conform to the complex geometrical shapes that represent the various subsurface features of interest is non-trivial. This pre-processing step, which involves generating meshes that effectively depict subsurface features, imposes a considerable computational burden and requires frequent analyst intervention (Annavarapu (2013)).