Solving fluid flow in discontinuous heterogeneous porous media and multi-layer strata with interpretable physics-encoded finite element network

Physics-informed neural networks (PINNs) have prevailed as differentiable simulators to investigate flow in porous media. Despite recent progress PINNs have achieved, practical geotechnical scenarios cannot be readily simulated because conventional PINNs fail in discontinuous heterogeneous porous media or multi-layer strata when labeled data are missing. This work aims to develop a universal network structure to encode the mass continuity equation and Darcy’s law without labeled data. The finite element approximation, which can decompose a complex heterogeneous domain into simpler ones, is adopted to build the differentiable network. Without conventional DNNs, physics-encoded finite element network (PEFEN) can avoid spectral bias and learn high-frequency functions with sharp/steep gradients. PEFEN rigorously encodes Dirichlet and Neumann boundary conditions without training. Benefiting from its discretized formulation, the discontinuous heterogeneous hydraulic conductivity is readily embedded into the network. Three typical cases are reproduced to corroborate PEFEN’s superior performance over conventional PINNs and the PINN with mixed formulation. PEFEN is sparse and demonstrated to be capable of dealing with heterogeneity with much fewer training iterations (less than 1/30) than the improved PINN with mixed formulation. Thus, PEFEN saves energy and contributes to low-carbon AI for science. The last two cases focus on common geotechnical settings of impermeable sheet pile in single-layer and multi-layer strata. PEFEN solves these cases with high accuracy, circumventing costly labeled data, extra computational burden, and additional treatment. Thus, this study warrants the further development and application of PEFEN as a novel differentiable network in porous flow of practical geotechnical engineering.