# Variational Geometry-aware Neural Network based Method for Solving High-dimensional Diffeomorphic Mapping Problems

arXiv: https://arxiv.org/abs/2511.01911 · HTML: https://mlsanity.com/paper/2511.01911/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 69 references that are in the map.
- First posted on arXiv 31 Oct 2025.

## Builds on (most cited first, 19 of 19)

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- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method](https://mlsanity.com/paper/2110.15832/) (2021)
- [Landmark and Intensity Based Registration with Large Deformations via Quasi-conformal Maps](https://mlsanity.com/paper/1310.1710/) (2013)
- [TEMPO: Feature-Endowed Teichm"uller Extremal Mappings of Point Clouds](https://mlsanity.com/paper/1511.06624/) (2015)
- [Topology-Preserving 3D Image Segmentation Based On Hyperelastic Regularization](https://mlsanity.com/paper/2103.16768/) (2021)
- [Fast Disk Conformal Parameterization of Simply-connected Open Surfaces](https://mlsanity.com/paper/1408.6974/) (2014)
- [Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing](https://mlsanity.com/paper/1508.07569/) (2015)
- [Spherical Density-Equalizing Map for Genus-0 Closed Surfaces](https://mlsanity.com/paper/2401.11795/) (2024)
- [Shape analysis via inconsistent surface registration](https://mlsanity.com/paper/2003.01357/) (2020)
- [Parallelizable global conformal parameterization of simply-connected surfaces via partial welding](https://mlsanity.com/paper/1903.12359/) (2019)
- [A unifying framework for n-dimensional quasi-conformal mappings](https://mlsanity.com/paper/2110.10437/) (2021)
- [Tooth morphometry using quasi-conformal theory](https://mlsanity.com/paper/1901.01651/) (2019)
- [Bijective Density-Equalizing Quasiconformal Map for Multiply-Connected Open Surfaces](https://mlsanity.com/paper/2308.05579/) (2023)
- [A Linear Formulation for Disk Conformal Parameterization of Simply-Connected Open Surfaces](https://mlsanity.com/paper/1508.00396/) (2015)
- [Efficient Feature-based Image Registration by Mapping Sparsified Surfaces](https://mlsanity.com/paper/1605.06215/) (2016)
- [Fast Spherical Quasiconformal Parameterization of Genus-0 Closed Surfaces with Application to Adaptive Remeshing](https://mlsanity.com/paper/1608.01140/) (2016)

## Abstract (arXiv)

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.