  We introduce a new class of numerical schemes which allow for low regularity
approximations to the expectation $ \mathbb{E}(|u_{k}(\tau, v^{\eta})|^2)$,
where $u_k$ denotes the $k$-th Fourier coefficient of the solution $u$ of the
dispersive equation and $ v^{\eta}(x) $ the associated random initial data.
This quantity plays an important role in physics, in particular in the study of
wave turbulence where one needs to adopt a statistical approach in order to
obtain deep insight into the generic long-time behaviour of solutions to
dispersive equations. Our new class of schemes is based on Wick's theorem and
Feynman diagrams together with a resonance based discretisation (see
arXiv:2005.01649) set in a more general context: we introduce a novel
combinatorial structure called paired decorated forests which are two decorated
trees whose decorations on the leaves come in pair. The character of the scheme
draws its inspiration from the treatment of singular stochastic partial
differential equations via Regularity Structures. In contrast to classical
approaches, we do not discretize the PDE itself, but rather its expectation.
This allows us to heavily exploit the optimal resonance structure and
underlying gain in regularity on the finite dimensional (discrete) level.
