A combination of physics-informed neural networks with the fixed-stress splitting iteration for solving Biot's model

Biot's consolidation model in poroelasticity describes the interaction between the fluid and the deformable porous structure. Based on the fixed-stress splitting iterative method proposed by Mikelic et al. (Computat Geosci, 2013), we present a network approach to solve Biot's consolidation model using physics-informed neural networks (PINNs).Two independent and small neural networks are used to solve the displacement and pressure variables separately. Accordingly, separate loss functions are proposed, and the fixed stress splitting iterative algorithm is used to couple these variables. Error analysis is provided to support the capability of the proposed fixed-stress splitting-based PINNs (FS-PINNs).Several numerical experiments are performed to evaluate the effectiveness and accuracy of our approach, including the pure Dirichlet problem, the mixed partial Neumann and partial Dirichlet problem, and the Barry-Mercer's problem. The performance of FS-PINNs is superior to traditional PINNs, demonstrating the effectiveness of our approach.Our study highlights the successful application of PINNs with the fixed-stress splitting iterative method to tackle Biot's model. The ability to use independent neural networks for displacement and pressure offers computational advantages while maintaining accuracy. The proposed approach shows promising potential for solving other similar geoscientific problems.