Physics informed neural networks based on mixed pressure-velocity formulation for flow in heterogeneous aquifers

Physics-Informed Neural Networks (PINNs) have emerged as a promising paradigm for solving problems governed by partial differential equations (PDEs) using the flexibility and generalization capability of deep learning. By embedding the governing physical laws directly into the training process, PINNs can approximate complex physical systems even when limited or no observational data are available. However, their performance and convergence can deteriorate significantly in domains characterized by high heterogeneity or discontinuities in material properties. In particular, standard PINN formulations tend to enforce implicit continuity in the hydraulic conductivity field, which can lead to inaccurate representations of physical processes in heterogeneous porous media.

This study introduces a novel and robust PINN framework for modelling transient fluid flow in heterogeneous porous media, with specific emphasis on accurately handling discontinuities in the hydraulic conductivity field. The proposed approach is based on a mixed formulation of the governing flow equations, in which the pressure and velocity fields are represented by independent neural networks. This structural separation eliminates the need to compute spatial derivatives of discontinuous or non-differentiable quantities during the evaluation of the loss function. As a result, the method achieves a more stable and accurate application of automatic differentiation while maintaining strong adherence to the underlying physical principles.

Furthermore, to address the high computational cost typically associated with training PINNs, a discrete-time mixed formulation is developed. By discretizing the temporal domain, this approach reduces the dimensionality of the problem, leading to substantial savings in both memory usage and training time. Despite these efficiency gains, the discrete-time PINN retains a high level of accuracy and fidelity in predicting transient flow dynamics in heterogeneous domains.

Comprehensive testing on various scenarios of unconfined aquifers demonstrate that the proposed implementation outperforms standard PINN approaches when applied to porous media with strong contrasts in hydraulic conductivity. The results obtained from the different PINNs techniques have been compared against the results from finite element software COMSOL to analyze their performance.

Overall, the mixed formulation PINN frameworks are computationally more efficient, and produce results with improved accuracy compared to the standard PINNs technique for simulating fluid flow in complex porous media systems, representing a significant step forward in the application of deep learning to subsurface modelling.