The use of neural networks has shown significant potential to reduce the computational costs associated with the dynamics of industrial computational fluids. Weak adversarial networks (WAN) leverage weak solution theory to transform the problem of solving PDEs into a Min-Max optimization problem, which is then solved by training a generative adversarial network. Although this method has been successfully applied to two-dimensional (2D) Navier-Stokes (NS) equations, previous work says nothing about the NS equation in porous media. In this study, we first leverage the stream function to introduce the biharmonic formulation of NS in porous media. Then, we extend the WAN framework to solve NS equations in porous media (WAN2DNS-PM) and provide free surface flow as a numerical experiment. Our results demonstrate the stability and accuracy of the proposed method, highlighting its advantages over the traditional Physic-Informed Neural Networks (PINNs) algorithm, particularly for problems lacking strong solutions. This work contributes to the growing research on AI-driven numerical methods for complex fluid dynamics problems, offering a promising approach for industrial applications.