We will discuss a weak universality phenomenon in the context of two-dimensional fractional nonlinear wave equations. For a sequence of Hamiltonians of high-degree potentials scaling to the fractional $Φ_2^4$, we will present a sufficient and almost necessary criteria for the convergence of invariant measures to the fractional $Φ_2^4$. Then we will discuss the convergence result for the sequence of associated wave dynamics to the (renormalized) cubic wave equation. This extends a result of Gubinelli-Koch-Oh to a situation where we do not have any local Cauchy theory with highly supercritical nonlinearities. This is a joint work with Chenmin Sun and Weijun Xu.