Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Generalizable Motion Planning via Operator Learning
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Proposes a Planning Neural Operator (PNO) that learns the Eikonal PDE solution operator from cost maps to value functions, enabling zero-shot super-resolution, cross-environment generalization, and use as an epsilon-consistent A* heuristic.
Problem
Motion planning value function prediction is costly with classical numerical Eikonal solvers and PINN-based neural planners require retraining per environment; learned value functions also may generalize poorly across resolutions, environments, and goals.
Approach
Recast value function approximation as learning a single neural operator from cost-function space to value-function space via the Eikonal PDE; extend FNO with a hard-encoded smoothed obstacle indicator (from an FNO-predicted SDF) in Fourier layers and a non-negative-weight Deepnorm projection that enforces the triangle inequality for goal generalization; use the output as an eroded epsilon-consist
Key results
- PNO with PINN loss achieves relative L2 error 0.0872 at 1024^2 test resolution trained on 64^2 data, and about 0.184 on MovingAI city maps versus 0.43 for DAFNO, with ~10x speedup over FMM at 1024^2 (8.3 ms vs 104.6 ms).
- As an A* heuristic with erosion on MovingAI maps, PNO reduces expanded nodes by about 31-33% versus the Euclidean heuristic (e.g., 2024 vs 3310 at 256^2) with near-optimal path length.
- On 4-DOF manipulator C-space, PNO achieves 0.027 training and 0.041 test relative L2 error.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add obstacle geometry encoding to FNO (ablation) | PNO · MovingAI synthetic obstacle maps | relative L2 error: 0.1996 -> 0.0985 | Helped |
| add obstacle geometry encoding to FNO (ablation) | PNO · MovingAI real-world city maps 1024² | relative L2 error: 0.7692 -> 0.4315 | Helped |
| add Deepnorm layer to DAFNO (ablation) | PNO · MovingAI real-world city maps 1024² | relative L2 error: 0.4315 -> 0.2034 | Helped |
| add Deepnorm layer to DAFNO (ablation) | PNO · MovingAI synthetic obstacle maps 64² | relative L2 error: 0.0985 -> 0.1136 | Hurt |
| add PINN loss (ablation) | PNO · MovingAI synthetic maps 64² | relative L2 error: 0.1136 -> 0.0698 | Helped |
| add PINN loss (ablation) | PNO · MovingAI real-world city maps 1024² | relative L2 error: 0.2034 -> 0.1842 | Helped |
| add obstacle erosion to PNO heuristic (ablation) | A* with PNO heuristic · MovingAI 2D city 256² | path length: 144.88 -> 144.05 | Helped |
| add obstacle erosion to PNO heuristic (ablation) | A* with PNO heuristic · MovingAI 2D city 256² | epsilon suboptimality estimate: 1.005 -> 1.000 | Helped |
| use PNO heuristic with erosion instead of Euclidean norm | A* · MovingAI 2D city 256² | nodes expanded: 3310 -> 2024 | Helped |
Surprising findings
- Without erosion, the PNO heuristic yields suboptimal paths (epsilon suboptimality up to 1.015), so erosion is needed for near-optimality.
Limitations
- Heuristic path lengths can be suboptimal without erosion
- Only uniform step costs considered
- Relative test error on iGibson (0.19) is high compared with training (0.08)
Experimental setup
Datasets: Grid-world dataset (VIN) MovingAI 2D city maps iGibson building dataset HouseExpo synthetic 64x64 obstacle dataset
Benchmarks: MovingAI lab 2D city maps iGibson 3D scenes Grid-world 8^2/16^2/28^2 4-DOF manipulator C-space
Metrics: relative L2 error success rate computation time nodes expanded path length epsilon suboptimality
Baselines: FMM VIN IEF2D NTFields P-NTFields FNO DAFNO A* RRT RRT*
Builds on: Fourier neural operator Domain-Agnostic Fourier Neural Operator NTFields A*
Closest papers by meaning
- Path Planning using Neural A* Search (ICML 2021 · similarity 0.86)
- Motion Planning in Compressed Representation Spaces (ICML 2026 · similarity 0.85)
- Potential Based Diffusion Motion Planning (ICML 2024 · similarity 0.84)
- Physics-informed Temporal Difference Metric Learning for Robot Motion Planning (ICLR 2025 · similarity 0.83)
- Scaling Value Iteration Networks to 5000 Layers for Extreme Long-Term Planning (ICML 2025 · similarity 0.83)
- Adaptive Coordinate Transforms for Neural Operators (NeurIPS 2026 · similarity 0.82)
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