Weak adversarial networks for solving forward and inverse problems involving 2D incompressible Navier–Stokes equations

In this paper, we study the weak adversarial network (WAN) (Zang et al. in J Comput Phys 411:109409, 2020) method for solving Navier–Stokes (NS) equation numerically, including both forward and inverse problems. Firstly, the NS equation is converted to the biharmonic formulation leveraging the stream function, then the biharmonic formulation equation contributes to a test function as an operator. Furthermore, the underlying NS equation is discretized by using two neural networks based on weak formulation. Finally, numerical results for some experiments are presented, this method shows advantages of stability and accuracy compared with other algorithms, especially this method is useful for problems which have no strong solutions.