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

2025/10/31 by Li, Zhiwen, Ho, Cheuk Hin, Lui, Lok Ming
#Artificial Intelligence (cs.AI) #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2511.01911

Abstract

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.

Citations

• Hybrid Finite Difference with the Physics-informed Neural Network for solving PDE in complex geometries
• Topology-Preserving 3D Image Segmentation Based On Hyperelastic Regularization
• PyTorch: An Imperative Style, High-Performance Deep Learning Library
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems
• Adam: A Method for Stochastic Optimization
• Landmark and Intensity Based Registration with Large Deformations via\n Quasi-conformal Maps

Related

• A Deep Learning Framework for Diffeomorphic Mapping Problems via Quasi-conformal Geometry applied to Imaging
• NODEO: A Neural Ordinary Differential Equation Based Optimization Framework for Deformable Image Registration
• Deform-GAN:An Unsupervised Learning Model for Deformable Registration
• A Symmetric Dynamic Learning Framework for Diffeomorphic Medical Image Registration
• Fast Symmetric Diffeomorphic Image Registration with Convolutional Neural Networks