A novel approach called the Variational Geometry-aware Neural Network has been introduced to address the challenges of high-dimensional diffeomorphic mapping problems. This method employs a mesh-free learning framework that integrates variational principles with quasi-conformal theory, enabling it to produce accurate and bijective mappings. A key feature of this approach is its ability to maintain control over the quality of deformations, which is critical in high-dimensional settings. The method's effectiveness has been positively recognized, highlighting its potential to advance solutions in this complex domain. Developed and shared via arXiv, this technique reflects ongoing research efforts to improve computational methods for diffeomorphic mapping. Its design specifically targets the difficulties posed by dimensionality, offering a promising tool for applications requiring precise geometric transformations. This development aligns with recent trends in leveraging neural networks for sophisticated mathematical and geometric challenges.