Gallium Nitride High Electron Mobility Transistors (GaN-on-Si HEMTs) on silicon substrates exhibit significant potential in the field of radio-frequency devices due to its high frequency, high-speed, and low-cost advantages. Nevertheless, the thermal stress caused by the mismatch in lattice parameters and the coefficient of thermal expansion between the Si substrate and GaN epitaxial layer constrains the device performance. Due to the different temperature distributions in different scenarios, there is an urgent need for a model that can balance the contradiction between epitaxial growth and device operation for the joint design of GaN-on-Si HEMTs epitaxial structures. This work presents thermal stress formulas specifically designed for GaN epitaxial growth and device operation, based on Townsend’s theory of multilayer film curvature and stress. Furthermore, our work innovatively proposes a multi-scenario coupled GaN-on-Si HEMTs epitaxial structure design model based on multi-objective optimization. The model represents a combined objective function incorporating thermal stress experienced during device operation and epitaxial growth, as well as the total thermal resistance and bulk thermal conductivity of the epitaxial layer. Through simulation validation, the model has been proved to effectively calculate the optimized epitaxial structure parameters across multiple scenarios to meet diverse requirements. And it can effectively reduce the operating temperature. This model can accelerate the process of GaN epitaxial design and provide an effective method for enhancing the performance of GaN-on-Si HEMTs.