A mixed pressure-velocity formulation to model flow in heterogeneous porous media with physics-informed neural networks

François Lehmann, Marwan Fahs*, Ali Alhubail, Hussein Hoteit

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Current implementations of Physics Informed Neural Networks (PINNs) can experience convergence problems in simulating fluid flow in porous media with highly heterogeneous domains. The objective of this work is to develop an accurate implementation of PINNs to model fluid flow in heterogeneous porous media that avoids alternative approaches in the literature that tend to impose unphysical continuity of the hydraulic conductivity field. The proposed implementation is inspired by the mixed formulation of the governing equations, which separates the mass continuity equation and Darcy's law. This methodology has been used with the mixed finite element method and is known to provide accurate pressure and velocity fields in highly heterogeneous domains. In this work, a similar methodology is applied to PINNs, where the training loss function is based on the decoupled continuity equation and Darcy's law. The separation of the continuity equation and Darcy's law allows for calculating the residual term in the learning loss function without evaluating the spatial derivatives of the discontinuous hydraulic conductivity and associated non-differentiable functions. This approach provides more accurate automatic differentiation of the neural networks for pressure and velocity fields. The new implementation of PINNs can be applied to simulate flow in porous media, regardless of the type and level of heterogeneity with a discontinuous hydraulic conductivity distribution. A variety of structures of the new implementation are investigated. Numerical experiments show that the structure based on different neural networks for the pressure and each component of the velocity field provides the highest accuracy with equivalent training time as other structures. The proposed methodology presents a novel approach to broaden the scope of PINNs in modeling fluid flow in porous media, while preserving the precise representation of the domain's discontinuous features.

Original languageEnglish (US)
Article number104564
JournalAdvances in Water Resources
StatePublished - Nov 2023


  • Automatic differentiation
  • Flow in porous media
  • Heterogeneous domains
  • Mixed formulation
  • non-differentiable functions
  • Physics Informed Neural Networks (PINNs)

ASJC Scopus subject areas

  • Water Science and Technology


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