A Neural Network Framework for Geodesic-Like Curve Computation on Parametric Surfaces

A new computational framework applies deep learning to compute geodesic-like curves on parametric surfaces, a concept first introduced by Chen in 2010 for estimating shortest paths. The approach uses Physics-Informed Neural Networks to handle single surfaces and complex multi-surface systems with C^0 or higher continuity, as well as surfaces of revolution. The framework, detailed in a paper posted to arXiv on June 17, 2026, addresses a longstanding gap in the numerical computation of geodesic-like curves. While Chen established the theoretical convergence of these curves as approximations of true geodesics in 2010, an efficient numerical implementation had remained undeveloped [1][2]. The new method constructs a deep fully connected neural network to predict interior control points for a B-spline curve, minimizing an energy functional to find the minimal geodesic-like curve connecting two points on a regular parametric surface [4]. By combining the theoretical convergence properties of geodesic-like curves with the optimization capability of neural networks, the approach avoids directly solving the geodesic differential equations, which can suffer from computational instability and initialization dependence [3][4]. The framework is designed to work on surfaces defined by a parametrization over a rectangular domain, and it extends robustly to multi-surface systems with C^0 or higher continuity and surfaces of revolution [2][4]. Geodesic computation on surfaces has broader relevance in fields that rely on shape analysis. In computational anatomy, for instance, flows between coordinate systems are constrained to be geodesic flows satisfying the principle of least action, with diffeomorphic mappings used to study anatomical shape variability [6]. Accurate geodesic distances are also foundational in computer vision tasks such as image segmentation, where contours extracted from images can be used to reconstruct 3D geometries [8]. Other recent neural network methods have targeted geodesic distance estimation on discretized polygonal meshes. One such approach uses a local neural network-based solver within a dynamic programming scheme, processing coordinates and distance values of neighboring vertices to approximate the distance function at a target point [5]. That work noted that exact geodesic distances restricted to a polygon are at most second-order accurate, motivating higher-order methods [5]. The new PINN-based framework operates directly on parametric surfaces rather than mesh approximations, offering a complementary strategy for geodesic estimation.