(untitled) (Advances in Applied Mathematics and Mechanics, 2025)
In this paper, we consider the dual continuum fluid filtration problem. The mathematical model is described by a coupled system of partial differential equations for two pressure fields. We propose a novel PINN-based partial learning approach to solve this problem. We employ a partially explicit time scheme, where the implicit scheme is applied to a highly permeable continuum, and the explicit scheme is used for the low permeable continuum. We utilize discrete-time physics-informed neural networks (PINNs) to compute the implicit part of the problem that is challenging to solve directly. By leveraging the slow variations in the low permeable continuum and the robustness of PINNs to data noise, we decouple the equations and solve them sequentially. At each time layer, we first solve the implicit part and then treat it as a known function to solve the explicit part. The explicit part is spatially discretized using the finite element method with standard linear basis functions. We utilize transfer learning to accelerate the computation of the implicit part using the parameters from the previous time layer as the initial parameters for training the current time layer. To test the proposed method, we consider two two-dimensional model problems. For each problem, the method achieved good accuracy, with transfer learning significantly speeding up the computation.