Aerodynamic shape optimization (ASO) is a key technique in aerodynamic designs, aimed at enhancing an object’s physical performance while adhering to specific constraints. Traditional parameterization methods for ASO often require substantial manual tuning and are only limited to surface deformations. This paper introduces the Deep Geometric Mapping (DeepGeo) model, a fully automatic neural-network-based parameterization method for complex geometries. DeepGeo utilizes its universal approximation capability to provide large shape deformation freedom with global shape smoothness, while achieving effective optimization in high-dimensional design spaces. Additionally, DeepGeo integrates volumetric mesh deformation, simplifying the ASO pipeline. By eliminating the need for extensive datasets and hyperparameter tuning, DeepGeo significantly reduces implementation complexity and cost. Multiple case studies using the same parameterization settings, including the two-dimensional circle-to-airfoil optimization, the three-dimensional CRM wing optimization, and the Blended-Wing-Body aircraft optimization, demonstrate DeepGeo’s effectiveness compared to state-of-the-art free-form deformation methods. This research highlights DeepGeo’s potential to automate ASO, making it more accessible and efficient.