We introduce Neural Navigation Functions (Neural-NF), a learned reactive navigation function capable of zero-shot transfer across unseen environment geometries. Neural-NF places data-driven adaptation within a structured elliptic planner, where the navigation objective is learned while planner structure is preserved by construction. Specifically, intrinsic Laplacian-derived features are mapped to local PDE coefficients, and solving the resulting boundary value problem produces a globally consistent value function on each target domain. For every admissible learned model, the resulting policy is collision-free, provides monotonic descent and a global minimum at the goal by construction. This admits a linearly-solvable optimal-control interpretation for any parameter setting. Empirically, Neural-NF achieves strong zero-shot transfer across diverse geometries and outperforms learned planners that directly predict the value function by up to a 5× improvement.
Diffusion-based motion planners have demonstrated strong performance in generating diverse and high-quality robot trajectories in cluttered environments with multiple feasible solutions. However, existing approaches typically operate on fixed-length waypoint sequences, making the learned model resolution-dependent, thereby preventing zero-shot generalization across resolutions. In this work, we propose Function-Space Diffusion for Motion Planning (FSD-MP), a diffusion-based motion planner that models trajectories as continuous functions and performs diffusion directly in function space, achieving discretization-invariant trajectory generation. We define a mode-wise forward process in the spectral domain, driven by Gaussian noise with a Matérn-type covariance, and parameterize the reverse process with a boundary-compatible Discrete Sine Transform-based Fourier Neural Operator (DST-FNO) that preserves start-goal constraints across resolutions. We evaluate FSD-MP on 2D point robot and 7-DoF Franka manipulator planning benchmarks. Our method achieves competitive planning performance at the training resolution and generalizes zero-shot across resolutions up to 16× higher, preserving consistent planning behavior without retraining. These results demonstrate that function-space diffusion provides an effective framework for discretization-invariant motion planning.
Recent progress in visual navigation has largely been driven by scale: end-to-end policies with hundreds of millions of parameters trained on billions of frames or large-scale simulated data. We ask how much of this scale a single task family actually requires, and what structure can substitute for it. We propose a decomposed navigation model in which operations with known closed-form structure, such as projective geometry, occupancy, and coordinate transforms, are computed analytically and serve as interfaces between three small learned modules: an egress predictor that grounds the episode goal as a local subgoal in the current view, a navigation predictor that estimates a goal-conditioned posterior over where trajectories travel, and an endpoint-pinned residual diffusion generator that samples trajectory shapes from this posterior. The system trains only 0.58M out of a total of 22.7M parameters, on 44k frames in under one GPU-hour, yet approaches the performance of state-of-the-art models on navigation tasks across 6060 point-goal episodes and 60 environments, while having 233x fewer trainable parameters, the lowest collision rate among all evaluated methods, and 50 Hz inference speed. The decomposition further transfers to no-goal exploration by retraining only the 123k-parameter egress head, and its failure modes under sensor corruption are transparent and analytically correctable.
This paper presents a learning-based approach for all-pairs motion planning, where the initial and goal states are allowed to be arbitrary points in a safe set. We construct smooth goal-conditioned neural ordinary differential equations (neural ODEs) via bi-Lipschitz diffeomorphisms. Theoretical results show that the proposed model can provide guarantees of global exponential stability and safety (safe set forward invariance) regardless of goal location. Moreover, explicit bounds on convergence rate, tracking error, and vector field magnitude are established. Our approach admits a tractable learning implementation using bi-Lipschitz neural networks and can incorporate demonstration data. We illustrate the effectiveness of the proposed method on a 2D corridor navigation task.