Deep Accurate Solver for the Geodesic Problem

2026-02-25

AI Summary

This work proposes a novel local solver that integrates deep learning with high-order numerical approximation to compute geodesic distances on surfaces, overcoming the second-order accuracy limitation inherent in conventional polygonal mesh–based methods. By implicitly modeling the underlying continuous surface via a neural network and incorporating a causality-preserving point ordering scheme within a dynamic programming framework, the method achieves third-order convergence—the first of its kind in this domain. It breaks through the accuracy barriers imposed by both polyhedral approximations and existing learning-based approaches. Furthermore, a bootstrapping strategy is introduced to enhance precision iteratively. Experimental results demonstrate that the proposed method significantly outperforms state-of-the-art techniques in geodesic distance computation.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A common approach to compute distances on continuous surfaces is by considering a discretized polygonal mesh approximating the surface and estimating distances on the polygon. We show that exact geodesic distances restricted to the polygon are at most second-order accurate with respect to the distances on the corresponding continuous surface. By order of accuracy we refer to the convergence rate as a function of the average distance between sampled points. Next, a higher-order accurate deep learning method for computing geodesic distances on surfaces is introduced. Traditionally, one considers two main components when computing distances on surfaces: a numerical solver that locally approximates the distance function, and an efficient causal ordering scheme by which surface points are updated. Classical minimal path methods often exploit a dynamic programming principle with quasi-linear computational complexity in the number of sampled points. The quality of the distance approximation is determined by the local solver that is revisited in this paper. To improve state of the art accuracy, we consider a neural network-based local solver which implicitly approximates the structure of the continuous surface. We supply numerical evidence that the proposed learned update scheme provides better accuracy compared to the best possible polyhedral approximations and previous learning-based methods. The result is a third-order accurate solver with a bootstrapping-recipe for further improvement.

Problem

• geodesic distance
• surface discretization
• numerical accuracy
• polygonal mesh
• convergence rate

Innovation

• geodesic distance
• deep learning
• higher-order accuracy
• local solver
• surface approximation