Reduced One-Dimensional Models for Wave Turbulence System

While studying many wave turbulence (WT) phenomena, owing to their inherent complexity, one frequently encounters the necessity of uncertainty quantification (UQ) in high dimension. The implementation of many existing algorithms for the UQ under these circumstances demands a vast amount of computation and, as a consequence, the straightforward approach tends to be infeasible or impractical. One effort to circumvent this obstacle in high dimension is to find an effective dimension reduction procedure for the probability distribution of the dynamical system model. One of the methodologies to do this is to replace the true Markovian model with a simple stochastic model that is significantly more amenable to the UQ than the underlying system. The procedure can be carried out via approximating the original equation for each Fourier mode by an independent and analytically tractable stochastic differential equation. In this work, we introduce a new approach for the so-called reduced-order model strategy within the context of the Majda–McLaughlin–Tabak model. Our framework makes use of a detailed analysis of the one-dimensional WT prototype to build a family of simplified models. Furthermore, the adaptive parameters are tuned without performing a direct numerical simulation of the true dynamical system model.