Learning Neural Time Fields on Randers Manifolds for Drift-Aware Motion Planning

Physics-informed neural time fields provide a self-supervised learning framework for robot motion planning. Existing Eikonal and Riemannian formulations primarily model isotropic or symmetric motion costs. This limits their ability to represent directional effects arising from environmental flows, gravity, and direction dependent dynamics. We introduce Randers Neural Time Fields (RaNTFields), a physics-informed neural framework for learning asymmetric anisotropic travel-time fields. RaNTFields extends the Eikonal formulation using a Randers metric that combines a configuration-dependent Riemannian metric with a directional one-form, derived from the corresponding Hamilton-Jacobi equation. We evaluate RaNTFields on 2D navigation under complex flows, terrain navigation, and 5-DoF robot arm motion generation under gravitational drift. In constant-flow environments, RaNTFields achieves a 0.997 correlation with the Ordered Upwind reference solution across start-goal pairs. In the 5-DoF arm, it follows the preferred gravity drift direction and consequently reduces the peak torque by $17.8\%$ relative to an Eikonal planner, with a $96\%$ success rate compared to $98\%$ for the baseline. These results demonstrate that RaNTFields can encode directional preferences arising from environmental flows and robot dynamics.