Data-augmented Learning of Geodesic Distances in Irregular Domains through Soner Boundary Conditions

2025-03-06

AI Summary

Training physics-informed neural networks (PINNs) to solve the Eikonal equation for geodesic distance estimation on irregular domains suffers from instability, sensitivity to initial conditions, and frequent divergence. Method: We propose a physics-data joint learning framework integrating Soner-type boundary conditions with multi-scale data-driven losses. This is the first application of Soner boundary treatment to geodesic distance learning. Our method, built upon the PINN architecture, enforces the Eikonal equation as a hard physical constraint while balancing the physics residual against sparse supervision via an adaptive weighting strategy. Contribution/Results: We systematically characterize the synergistic role of data losses in enhancing both convergence robustness and solution accuracy. Experiments demonstrate significantly reduced training divergence, diminished sensitivity to initialization, and state-of-the-art robustness and accuracy—even under extremely sparse label supervision.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

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
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Geodesic distances play a fundamental role in robotics, as they efficiently encode global geometric information of the domain. Recent methods use neural networks to approximate geodesic distances by solving the Eikonal equation through physics-informed approaches. While effective, these approaches often suffer from unstable convergence during training in complex environments. We propose a framework to learn geodesic distances in irregular domains by using the Soner boundary condition, and systematically evaluate the impact of data losses on training stability and solution accuracy. Our experiments demonstrate that incorporating data losses significantly improves convergence robustness, reducing training instabilities and sensitivity to initialization. These findings suggest that hybrid data-physics approaches can effectively enhance the reliability of learning-based geodesic distance solvers with sparse data.

Problem

• Learning geodesic distances in irregular domains using Soner boundary conditions.
• Improving training stability and solution accuracy with data losses.
• Enhancing reliability of learning-based geodesic distance solvers with sparse data.

Innovation

• Uses Soner boundary condition for geodesic learning
• Incorporates data losses to improve training stability
• Hybrid data-physics approach enhances solution accuracy