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Generalizable Motion Planning via Operator Learning

Sharath Matada, Luke Bhan, Yuanyuan Shi, Nikolay Atanasov

TL;DR

This work reframes motion planning as learning the solution operator of an Eikonal PDE, mapping environment cost functions to optimal value functions $V(\bm{x})$ via a neural operator. The authors introduce the Planning Neural Operator (PNO), a resolution-invariant architecture that encodes obstacle geometry and enforces goal-position generalization through a triangle-inequality projection layer, enabling zero-shot super-resolution and cross-environment planning (including 3D iGibson and 4-DOF manipulators). A theoretical result guarantees the existence of a neural-operator approximation $\hat{\Psi}$ to the Eikonal solution operator, with an error bound $\epsilon$, under viscosity-solution and compactness assumptions. Empirically, PNO achieves fast, accurate value-function predictions at resolutions far beyond training (e.g., $16\times$ training resolution), outperforms several baselines on grid-world and real-world maps, and can serve as an $\epsilon$-consistent neural heuristic to substantially reduce node expansions in $A^*$ while preserving near-optimal paths. These advances enable efficient, generalizable motion planning in complex environments and offer practical impact for real-time robotic planning and navigation in changing scenes.

Abstract

In this work, we introduce a planning neural operator (PNO) for predicting the value function of a motion planning problem. We recast value function approximation as learning a single operator from the cost function space to the value function space, which is defined by an Eikonal partial differential equation (PDE). Therefore, our PNO model, despite being trained with a finite number of samples at coarse resolution, inherits the zero-shot super-resolution property of neural operators. We demonstrate accurate value function approximation at $16\times$ the training resolution on the MovingAI lab's 2D city dataset, compare with state-of-the-art neural value function predictors on 3D scenes from the iGibson building dataset and showcase optimal planning with 4-DOF robotic manipulators. Lastly, we investigate employing the value function output of PNO as a heuristic function to accelerate motion planning. We show theoretically that the PNO heuristic is $ε$-consistent by introducing an inductive bias layer that guarantees our value functions satisfy the triangle inequality. With our heuristic, we achieve a $30\%$ decrease in nodes visited while obtaining near optimal path lengths on the MovingAI lab 2D city dataset, compared to classical planning methods ($A^\ast$, $RRT^\ast$).

Generalizable Motion Planning via Operator Learning

TL;DR

This work reframes motion planning as learning the solution operator of an Eikonal PDE, mapping environment cost functions to optimal value functions via a neural operator. The authors introduce the Planning Neural Operator (PNO), a resolution-invariant architecture that encodes obstacle geometry and enforces goal-position generalization through a triangle-inequality projection layer, enabling zero-shot super-resolution and cross-environment planning (including 3D iGibson and 4-DOF manipulators). A theoretical result guarantees the existence of a neural-operator approximation to the Eikonal solution operator, with an error bound , under viscosity-solution and compactness assumptions. Empirically, PNO achieves fast, accurate value-function predictions at resolutions far beyond training (e.g., training resolution), outperforms several baselines on grid-world and real-world maps, and can serve as an -consistent neural heuristic to substantially reduce node expansions in while preserving near-optimal paths. These advances enable efficient, generalizable motion planning in complex environments and offer practical impact for real-time robotic planning and navigation in changing scenes.

Abstract

In this work, we introduce a planning neural operator (PNO) for predicting the value function of a motion planning problem. We recast value function approximation as learning a single operator from the cost function space to the value function space, which is defined by an Eikonal partial differential equation (PDE). Therefore, our PNO model, despite being trained with a finite number of samples at coarse resolution, inherits the zero-shot super-resolution property of neural operators. We demonstrate accurate value function approximation at the training resolution on the MovingAI lab's 2D city dataset, compare with state-of-the-art neural value function predictors on 3D scenes from the iGibson building dataset and showcase optimal planning with 4-DOF robotic manipulators. Lastly, we investigate employing the value function output of PNO as a heuristic function to accelerate motion planning. We show theoretically that the PNO heuristic is -consistent by introducing an inductive bias layer that guarantees our value functions satisfy the triangle inequality. With our heuristic, we achieve a decrease in nodes visited while obtaining near optimal path lengths on the MovingAI lab 2D city dataset, compared to classical planning methods (, ).

Paper Structure

This paper contains 27 sections, 5 theorems, 30 equations, 11 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Eikonal PDE formulation of the motion planning problem
  3. Motion planning as an optimal control problem
  4. Eikonal equation for optimal motion planning
  5. Properties and universal approximation of the Eikonal solution operator
  6. Designing Neural Operators for Eikonal PDEs
  7. Review of FNO and DAFNO
  8. PNO architecture: generalizing across environments
  9. PNO architecture: generalizing across goal positions
  10. Learning motion planning value functions
  11. Learned operator value functions as neural heuristics
  12. Conclusions, Limitations, and Future Work
  13. Proofs of Lemmas and Theorems
  14. Proof of Theorem \ref{['thm:nopExistence']}
  15. Proof of Lemma \ref{['lemma:epConsistent']}
  16. ...and 12 more sections

Key Result

Theorem 1

(See Appendix appendix:proofofThm2 for proof) Let Assumptions assumption:properPolicy, assumption:continuity_boundedness hold and consider $\Psi:\mathcal{F}_c \to \mathcal{F}_v$ as the solution to eq:eikonalProblem. Then, for any $\epsilon > 0$, there exists a neural operator $\hat{\Psi}: \mathcal{F

Figures (11)

  • Figure 1: Example of super-resolution capabilities of PNO for motion planning on a map of NYC. The operator is trained on a dataset of resolution $64\times64$ and the examples shown here (resolutions $256\times 256$, $512\times 512$, and $1024 \times 1024$) were not seen during training. See Sec. \ref{['sec:learning']} for details.
  • Figure 2: PNO network architecture. The input to a PNO is a binary occupancy grid, which is transformed into a sign distance function (SDF) via an independently trained FNO. This, along with the original binary map is passed to a series of modified FNO layers which hard encode the obstacles. Finally, this result, along with the goal, is then fed to a projection layer (ensuring satisfaction of the triangle inequality) obtaining the final value function prediction.
  • Figure 3: (left) Comparison of 3D value function approximations for two iGibson environments (best $L_2$ PNO, sliced at $z=0$). (right) On top, we show example paths in the Bolton environment. On the bottom table, we present a aggregate quantitative comparison. See Appendix \ref{['appendix:iGibson']} for more examples.
  • Figure 4: Planning with PNO generated value functions for a 4-DOF manipulator around two obstacles. The top and bottom rows show two separate examples with end-effector trajectory snapshots demonstrating motion from a start (blue dot) to a goal configuration (green dot).
  • Figure 5: Path planning using various heuristics on the Moving AI Shanghai city map. Each example on the left showcases the $A^\ast$ planning under the corresponding heuristic given on the right. The nodes expanded are in orange, the start in red, the goal in green, and the path in blue.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Definition 1
  • Lemma 1: $\epsilon$-consistency of neural heuristic
  • Lemma 2: Eroded heuristic is "more" consistent
  • proof
  • proof
  • proof
  • Theorem 2: Neural operator approximation theorem lanthaler2023nonlocal
  • Definition 2: Viscosity solution viscosityDef
  • Lemma 3