Physics-informed Evolutional Deep Neural Networks for Transient Pressure Diffusion in Heterogeneous Porous Media

Transient Darcy flow in heterogeneous porous media requires accurate pressure prediction and Darcy velocity reconstruction across space and time. In this work, we investigate an evolutional deep neural network (EDNN) framework for transient Darcy problems in one, two, and three spatial dimensions. Unlike space-time physics-informed neural networks that approximate the full solution field simultaneously, EDNN represents the pressure field at each time level and advances it through standard time-marching schemes, including explicit Euler, Crank-Nicolson, and fourth-order Runge-Kutta methods. From the viewpoint of numerical methods, this evolution-based formulation is natural for transient problems because it follows the intrinsic temporal structure of the governing equation. We evaluate the framework on benchmark cases with constant coefficients, smoothly varying permeability and storage coefficients, and low-permeability or discontinuous structures. The results show that EDNN provides accurate pressure prediction and reliable velocity reconstruction in smooth settings, and generally outperforms the space-time PINN baseline in higher-dimensional and heterogeneous cases. We further analyze the accuracy-efficiency trade-off of different time integrators. Higher-order integration improves performance for smooth variable-coefficient problems, whereas interface-dominated cases are mainly limited by spatial representation and pressure-gradient reconstruction. These findings demonstrate the potential of EDNN as a time-evolution-oriented neural framework for transient subsurface flow modeling.