Eccentric discs in rotational motion are commonly utilized in various technical fields, including gas turbine engines, flywheels, gears, and brakes. So, improving its critical angular velocity and frequency characteristics is a challenging issue for engineers. So, in this work for the first time, nonlinear vibrations and buckling analysis of the high-speed rotating eccentric disks using mathematical simulation and data-driven solutions are presented. One of the suggestions for improving its mechanical properties is considering metamaterials in the construction of the eccentric disk. Metamaterial is a novel synthetic material that has distinctive physical and mechanical capabilities that are unattainable in natural materials due to its well-designed structure. The properties of the eccentric disks are controlled by the amount of graphene and the degree of folding of graphene origami (GOri) across the thickness of the eccentric disks. These properties, such as Poisson's ratio, vary depending on the position and can be estimated using micromechanical models assisted by genetic programming (GP). Using von-Karman nonlinearity, transformed differential quadrature method (TDQM), and Newton's method the nonlinear governing equations are obtained and solved, respectively. The results show that when the radius ratio of the rotating eccentric disk increases by 8%, the critical point is decreased from 420 HZ to 370 HZ, a reduction of around 12 %. Another suggestion for improving its mechanical properties is controlling the geometrics and physics of the presented structure according to the designer's purposes.