In the framework of dynamical systems theory,the introduction of potential functions and scale parameters has led to the construction of three distinct types of scale-accepted topological upper metric mean dimensions and their corresponding measure-theoretic counterparts.This extension allows for a refined quantification of both local complexity and global statistical behavior of dynamical systems across varying scales.A systematic investigation has been conducted into the intrinsic relationships and structural differences among these dimensional definitions,revealing their hierarchical and complementary nature in characterizing dynamical complexity.The principal contribution of this work is the establishment of a variational principle for the scale-accepted upper metric mean dimension with potentials on subsets,which elucidates the extremal properties of such dimensions within the variational setting.This result provides a powerful analytical framework and novel theoretical tools for further exploration of complexity and geometric structures in dynamical systems with potentials.