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

TL;DR

This paper tackles the problem of learning geodesic distances in irregular domains using physics-informed neural networks. By decoupling boundary effects through the Soner boundary condition and combining physics losses with sparse data supervision, the authors demonstrate improved training stability and accuracy over physics-only approaches. They show that even 1–10 well-placed data points can match the performance of fully supervised methods while better enforcing boundary constraints. The results support a hybrid physics-data paradigm for reliable learning-based geodesic solvers, with practical implications for robust physics-informed neural motion planning in robotics.

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.

Figures (5)

• Figure 1: This figure demonstrates that even sparse data can significantly influence the learned geodesic distance functions. The color map represents the estimated geodesic distance to the source or goal point (red), and the arrows depict the gradient field. The left and right figures correspond to different placements of a single supervision point $x_{\mathrm{data}}$. While the right case converges to a poor minimum, the left case approaches the correct solution.
• Figure 2: The geodesic distance in flat euclidean space is computed in a square domain of side length $2$, and visualized through equidistant contour lines. The results arising from encoding boundaries through the speed model (left) and through the Soner condition (right) are illustrated in this figure. Note that the speed model variant "squishes" the distance close to the boundary, whereas the Soner condition allows for the exact solution.
• Figure 3: Canonical spaces designed for the ablation study on the effect of data losses on learning. From left to right: a convex domain, a non-convex domain, a non-simply connected domain. The shadowed area represents the free space, and the boundary is represented by the black lines. An example source point in the domain is shown in blue.
• Figure 4: The maximum absolute distance error at different dataset sizes are visualized in this figure. The statistics are shown for the non-convex (top) and the non-simply connected (bottom) environments over $12$ different trials. A general trend is observed of declining mean error with an increasing dataset size. The standard deviation of the errors at dataset sizes of $1$ and $10$ are considerably higher than at larger datasets.
• Figure 5: Visualization of the robustness to noise, using two different datasets with a fixed number of points ($10$), and the two extreme levels of noise ${\eta=[0, 0.25]}$. The source point is depicted in red; the points in the dataset are shown in light purple. Both trials converge to qualitatively similar solutions at all evaluated noise levels. The dataset (a) guides convergence to a poor minimum, while dataset (b) guides solutions to good approximations.