用神经网络实现无需固定分辨率的几何映射，高效处理空间变化场。

Many imaging problems require computing spatial transformations induced by spatially varying intensity, feature, or density fields. Canonical examples include distortion correction, deformable image registration, atlas-based segmentation, and deformation-driven image analysis. These tasks can be formulated as geometric mapping problems in which the transformation is constrained to preserve local structure, control boundary behavior, or regulate angular distortion. Such formulations typically lead to variational models, diffusion processes, or elliptic partial differential equations. However, repeatedly solving high-resolution systems becomes computationally expensive when the underlying parameter fields vary across instances. In this work, we propose a resolution-free neural surrogate for geometric parameterization and mapping problems. Given a spatially varying parameter field $p:Ω\to\mathbb{R}^m$ and query locations $\{x_i\}_{i=1}^N\subsetΩ$, the model predicts mapped locations $\{u(x_i)\}_{i=1}^N$ on arbitrary structured or unstructured point sets. To avoid dependence on a fixed grid, we use a multi-resolution geometric encoding strategy that conditions the network on coordinate-augmented samples of the parameter field. The model is trained without labeled solution data by enforcing geometry-aware constraints derived from variational energies, diffusion-based density equalization, and quasi-conformal theory. Experimental results on quasi-conformal mapping and density-equalizing mapping problems are presented to demonstrate the effectiveness of our proposed method.