Traditional methods for high-dimensional diffeomorphic mapping often struggle with the curse of dimensionality. We propose a mesh-free learning framework designed for $n$-dimensional mapping problems, seamlessly combining variational principles with quasi-conformal theory. Our approach ensures accurate, bijective mappings by regulating conformality distortion and volume distortion, enabling robust control over deformation quality. The framework is inherently compatible with gradient-based optimization and neural network architectures, making it highly flexible and scalable to higher-dimensional settings. Numerical experiments on both synthetic and real-world medical image data validate the accuracy, robustness, and effectiveness of the proposed method in complex registration scenarios.

翻译：传统的高维微分同胚映射方法常受维度灾难困扰。我们提出一种面向$n$维映射问题的无网格学习框架，将变分原理与拟共形理论有机结合。该方法通过调控共形畸变与体积畸变，确保精确的双射映射，实现对变形质量的鲁棒控制。该框架天然兼容基于梯度的优化与神经网络架构，具有高度灵活性，并可扩展至更高维场景。在合成数据与真实医学影像数据上的数值实验验证了所提方法在复杂配准任务中的准确性、鲁棒性与有效性。