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.　©　2024,　The　Author(s)　under　exclusive　licence　to　Sociedade　Brasileira　de　Matemática　Aplicada　e　Computacional.