We have developed a robust method of building at least metastable atomistic models of a bicontinuous cubic, lyotropic liquid crystal polymer membrane. Our approach helps overcome the longstanding problem of modeling bicontinuous cubic structures at the atomic level. To achieve this, we use a novel combination of mathematical constraints, CG modeling, and atomistic simulations to achieve atomistic systems stable for hundreds of nanoseconds. Furthermore, the atomistic model can be used to design and modify the LLC polymer membrane reducing the time and cost of development of these membranes for their applications.