Finding geodesics with the Deep Ritz method

TL;DR

The paper addresses finding geodesics between fixed endpoints on parameterized surfaces by formulating the problem as a variational boundary-value problem and solving it with the Deep Ritz method (DRM). By parameterizing the trajectory with a neural network and minimizing the discretized energy $E(\boldsymbol \theta(t))$ while enforcing Dirichlet boundaries automatically, the authors avoid shooting methods and demonstrate results across path planning, optics, solid mechanics, and latent-space interpolation. They derive the geodesic equations through the metric $g_{jk}(\boldsymbol \theta)$ and Christoffel symbols $\Gamma_{ijk}$, and show DRM optimizes a nonlinear system driven by the variational structure. The work highlights architectural strategies (e.g., Fourier feature embeddings) to overcome local minima and suggests that DRM can offer a competitive, scalable alternative for geodesic computations in SciML, with broad implications for trajectory optimization and generative modeling.

Abstract

Geodesic problems involve computing trajectories between prescribed initial and final states to minimize a user-defined measure of distance, cost, or energy. They arise throughout physics and engineering -- for instance, in determining optimal paths through complex environments, modeling light propagation in refractive media, and the study of spacetime trajectories in control theory and general relativity. Despite their ubiquity, the scientific machine learning (SciML) community has given relatively little attention to investigating its methods in the context of these problems. In this work, we argue that given their simple geometry, variational structure, and natural nonlinearity, geodesic problems are particularly well-suited for the Deep Ritz method. We substantiate this claim with four numerical examples drawn from path planning, optics, solid mechanics, and generative modeling. Our goal is not to provide an exhaustive study of geodesic problems, but rather to identify a promising application of the Deep Ritz method and a fruitful direction for future SciML research.

Figures (11)

• Figure 1: Different types of terrain are obtained by varying the height and frequency of the hills with $h$ and $f$ respectively. Intuitively, taller and more frequent hills necessitate circuitous paths.
• Figure 2: When the landscape is mellow, approximately straight line paths are minimal distance, despite crossing over some foothills.
• Figure 3: More severe terrain requires staying in valleys, as the cost of climbing uphill to move in a straight line overshadows the cost of wandering.
• Figure 4: Visualizing the centerline of the helix-shaped wave guide (left) and the helical tube in which the wave speed is large (right). Outside of the tube, the refractive index is large, indicating longer travel times.
• Figure 5: When using embedded Fourier features, the optimizer manages to avoid the straight line local minimum, finding a winding path through the helical region of large wave speeds, which is shown in purple.