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.

翻译：我们引入了一类新型数值格式，用于在低正则条件下逼近期望值 $\mathbb{E}(|u_{k}(\tau, v^{\eta})|^2)$，其中 $u_k$ 表示色散方程解 $u$ 的第 $k$ 个傅里叶系数，$v^{\eta}(x)$ 为相应的随机初始数据。该量在物理学中具有重要作用，尤其在波湍流研究中，需要通过统计方法深入理解色散方程解的长时间普遍行为。这类新格式基于Wick定理与费曼图，并结合共振离散化方法（参见arXiv:2005.01649），在更广义的框架下提出了一种新型组合结构——配对角叶森林：即两棵装饰树，其叶节点装饰成对出现。该格式的核心思想借鉴了正则结构理论对奇异随机偏微分方程的处理方式。与经典方法不同，我们不对PDE本身进行离散化，而是直接离散化其数学期望。这使得我们能够在有限维（离散）层面上，充分挖掘最优共振结构及潜在的规则性增益。