Generalizable Motion Planning via Operator Learning

Authors: Sharath Matada, Luke Bhan, Yuanyuan Shi, Nikolay Atanasov
Primary area: applications to robotics, autonomy, planning
Keywords: Motion planning, operator learning, value function approximation, Eikonal PDE

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× 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-joint 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 $\epsilon$-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$^\ast$).

Decision: accept (poster)
Reviewer scores: 5, 6, 6

Reviews and discussion:
<thread>
creation date:
1730587109829

thread_type:
official_review

<thread_object>
creator: reviewer_hzwb

<content>
summary:
This paper proposes a method, called Planning Neural Operator (PNO), for approximating the value function of a motion planning problem, viewed as the solution operator to the Eikonal PDE. The authors claim to achieve generalizable motion planning by using a resolution-invariant architecture, which encodes obstacle geometries as well as goal positions. This is done by leveraging the Fourier Neural Operator (FNO) and the Domain-Agnostic FNO (DAFNO). The paper also proposed a method for integrating the learned operator as a heuristic into the A* algorithm which achieves a 33% decrease in the number of explored nodes while maintaining epsilon-consistent optimality.

strengths:
- The paper, although math-heavy, was an enjoyable read. The authors did a very good job of presenting complex theoretical concepts simply while maintaining a good formalism.
- The authors provide a detailed literature review and clearly position the proposed approach within existing work  in traditional and neural motion planning.
- Experiments are conducted on multiple datasets, showcasing the performance of the method on different types of 2D and 3D environments of different resolutions. Results support most of the authors’ claims.

weaknesses:
- Several assumptions are made to justify the existence of the neural operator approximation. However, the reasoning behind these assumptions, and their implications in practical, real-world scenarios, are not thoroughly discussed.
- The model’s performance for robots with high degrees of freedom (DOF) such as manipulators are not explored. This raises questions about the applicability of the approach to more complex, higher-dimensional tasks common in robotic motion planning.
- While the paper mentions design decisions like encoding obstacle geometries and ensuring triangle inequality, the specific impact of these features is not thoroughly examined through ablation studies. It would be beneficial to quantitatively show how each architectural component contributes to the final performance, especially the modifications to the Fourier Neural Operator (FNO). Also, comparing the use of DeepNorm to other possible parametrizations could help justify the design choices even more.

questions:
- Could you provide more details on the intuition behind the assumptions required in the paper and how it relates to the practical application of your method in real-world scenarios?
- Have you tested PNO in real-world robotic setups beyond the synthetic and iGibson environments? If so, could you share results or insights on how PNO performs in tasks involving more complex real-world constraints or unstructured environments?
- How does the proposed method scale to tasks with significantly higher degrees of freedom (DOF) beyond the environments presented in the paper? Have you performed any scalability studies on more complex robotic systems?
- Could you provide more detailed ablation studies that show the contributions of specific components in your architecture, such as obstacle encoding, triangle inequality enforcement, and modified Fourier Neural Operator layers? This would clarify how each design choice impacts overall model performance.
- In table 1, do you have an explanation on why the success rate of your method particularly drops for the 16x16 maps ?
</content>
</thread_object>

</thread>
<thread>
creation date:
1730613778968

thread_type:
official_review

<thread_object>
creator: reviewer_6xqc

<content>
summary:
This paper introduces the novel use of neural operators to solve the Eikonal equation for motion planning in 2D and 3D environments. The proposed method takes an environment occupancy map, along with start and goal points, to predict the optimal value function for path planning in free space. By leveraging the generalization capabilities of neural operators, the authors demonstrate improved adaptability across different environments compared to conventional methods. Experimental comparisons with search-based methods and neural motion planners validate the effectiveness and computational efficiency of this approach.

strengths:
1. Innovative Use of Neural Operators: The authors adopt neural operators, which have shown superior generalization in PDE-solving applications, suggesting potential advancements in both performance and scalability.

2. Generalization Across Datasets: Experimental results indicate the method’s potential to generalize across diverse environments, which is a noted advantage over physics-informed neural networks (PINN) approaches.

3. Computational Efficiency: The approach demonstrates fast computational performance for small-resolution cases, making it potentially suitable for real-time applications.

weaknesses:
1. Scalability Constraints:
Since the method relies on grid-space convolutions, it faces intrinsic limitations in scaling to higher-dimensional spaces, a notable restriction for applications in manipulation tasks.

2. Dependence on Supervised Learning: 
 The method’s reliance on supervised learning demands ground truth PDE solutions, which may be impractical to obtain in complex environments.

3. Presentation Issues:
 - Notation Clarity: The pipeline figure uses the same symbol for the SDF operator and PNO, leading to ambiguity. It would be clearer to differentiate these in the figure.
- The neural network structure lacks clarity. The description mentions $F^{-1}(R(F()))$, but it may be more informative to explicitly state the intermediate layers, e.g., $L = F^{-1}(R(F()))$, with a clear distinction that the output should be denoted as $\Phi $ rather than $L_M $.

4. Experimental Comparisons:
 The paper currently compares against VIN and NTFields, which are established but not necessarily state-of-the-art. Incorporating comparisons with more advanced methods like Differentiable Spatial Planning using Transformers [1] and Progressive Learning for Physics-Informed Neural Motion Planning [2] would provide a more comprehensive benchmark.

[1] Differentiable Spatial Planning using Transformers

[2] Progressive Learning for Physics-Informed Neural Motion Planning

The original experiment struggled to fit the value function accurately, but adding the PINN loss has shown improvements. However, the current method has only been tested on environments with isolated cluttered obstacles. This makes it difficult to assess its ability to generalize to more complex scenarios, such as the maze-like Gibson environments, where optimal paths often require navigating through multiple turns.

questions:
Despite its limitations in scaling and reliance on ground truth training data, the proposed method represents a promising direction for motion planning through neural operators. I have the following questions to further explore its potential:

1. Pure PINN Loss: Could pure PINN loss be used to train the network instead of supervised learning? An analysis of its generalization performance could reveal insights into the potential for unsupervised or semi-supervised learning.

2. Testing on General Geometries: The method’s scalability could be further assessed by testing on general geometries, as seen in [1]. This could help to understand whether the approach can extend to higher-dimensional space planning.

3. Visualization of Results: It would be more effective to display contour lines of the value function instead of only color representations. Contour lines could better reveal any artifacts in the value function, providing clearer insights into the method's performance.

[1] Fourier Neural Operator with Learned Deformations for PDEs on General Geometries
</content>
</thread_object>

</thread>
<thread>
creation date:
1730953933548

thread_type:
official_review

<thread_object>
creator: reviewer_c6nw

<content>
summary:
This paper casts motion planning as a problem of learning a value function modelled by an Eikonal PDE. The authors propose to view the solution as a non-linear operator mapping costs to value function solution, and learn this using fourier neural operators. The paper proposes to adjust standard Fourier neural operators by (1) encoding the obstacles, given by an SDF, and (2) enforce triangle inequality is satisfied. The paper argues that the neural operator formulation enables the querying to be done in at a higher resolution than the resolution of 2D maps used in training.

I appreciate the novelty of casting motion planning into an operator learning problem, and the approach presented is solid. My main concern is a critical one: the simplicity of the experimental setups -- the evaluations were all conducted in 2d "grid-world"-like environments (point mass moving in 2d layouts e.g. Fig 4.), whereas other methods of learning how to plan generally evaluate on higher degrees of freedom tasks, such as robot manipulators of 6 or 7 DOF.

strengths:
See review above

weaknesses:
See review above

questions:
How well does the learned planner scale wrt the dimensions of the search space? As most of the citations come from the robotics literature, I would like to see motion planning beyond 2D or 3D toy examples. Consider evaluation on robot manipulators with 6 or 7 DOF.
</content>
</thread_object>

</thread>
<thread>
creation date:
1732649255428

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Overall response

comment:
We thank all reviewers for their helpful and insightful comments. We appreciate that the reviewers recognize the innovative use of operator learning as well as the theoretical foundation of our work. First, we address common questions and concerns of all reviewers and then we provide an individual response to each reviewer. We have attached two versions of the document. The main PDF contains the revisions, and the supplementary is the same document with the edits highlighted in $\color{red}\text{red}$ for convenience.

**Presentation of theoretical results**

We recognize that the mathematical formulation of the operator learning problem is challenging and have made the following edits to improve the presentation. 
1. We added comments about the feasibility of Assumption 1 and 2, emphasizing that both assumptions are achievable in practice.
2. As suggested by Reviewer 6Xqc, we improved the diagram overviewing the architecture of our model and added explanations for the mathematical formulation of the layers in our model to Section 3. We hope that this clarifies the exact structure of PNO as well as solidifies the reasoning for each component.  

**Ablation study and additional benchmark comparison**

We thank the reviewers for their suggestions to highlight the improvements in our architecture compared to the state of the art. We made the following additions.
1. We have altered the MovingAI city map section to highlight the advantages of
each component of our architecture in an ablation study format. In particular,
we showcase the importance of obstacle encoding in DAFNO and PNO over FNO
as well as the effect of the triangle inequality preservation in PNO as it achieves
much better generalization across different goals compared to DAFNO. These
results are presented in Table 2 with additional discussion in the exposition of
Section 4, ”Ablation Study: Moving AI City Maps”.
2. We added comparisons to P-NTFields [1] on the 3D iGibson environments. The idea behind P-NTFields is that during training the speed-field is slowly increased from $0.5$ to $\approx 1$ to eliminate local minima. As a result, as shown in Figure 3, P-NTFields indeed generates very nice paths. However, quantitatively, due to this training methodology, the value function captured by P-NTFields differs in magnitude from FMM because the FMM input velocity-field is a constant $1$. Despite this difference in magnitude, one can see that in the contour plots, FMM and P-NTFields largely match. We added a sentence in the paper to highlight this clarification.

**Scalability and application to manipulators**

To showcase the ability of our approach to handle more complex tasks, we introduced a new experiment applying PNO to a 4-DOF robot manipulator planning problem. In the 4-DOF problem, we demonstrate that PNO is able to compute value functions in $C-$space (Configuration space). We show visualizations of the value function predictions in Figure 8 in the Appendix and two examples of manipulator path executions in Figure 4.  

In future work, we plan to apply the PNO architecture to real-world robot manipulators with higher DOF but we emphasize that our focus in this work is on the foundation of the operator learning approach and its strengths. To extend our method to higher DOF manipulators, we recognize the challenge of using point-wise training data and thus will need to invoke different representations of the environment. However, this will introduce new challenges of ensuring obstacle avoidance and the satisfaction of the triangle inequality in latent space, leaving an exciting direction for future research.
 
**Overall impact and contributions:**

We briefly summarize the impact and contributions of the paper. This paper lays the groundwork for developing super-resolution value function approximation in continuous function space by introducing the first connection between neural operators and the Eikonal PDE in motion planning problems. We prove the existence of an arbitrarily good neural-operator approximation theoretically and highlight the architectural advantages of super-resolution operator-based motion planning across 2D, 3D, and 4D experiments. We then give an example of the utility of super-resolution value function approximation by employing the PNO value predictions as heuristics for motion planning algorithms, proving both $\epsilon-$consistency of the heuristic as well as showing improvement in the number of expanded nodes in $A^\ast$. The key advantage of the approach across all experiments is PNO's ability to *generalize* to new environments, while enabling super-fine discretization for accurate planning. Thus, this work opens a new perspective for designing neural motion planners that function across different environments while maintaining the high accuracy needed for motion planning. 

[1] R. Ni and A. H. Qureshi. Progressive learning for physics-informed neural motion planning. Robotics: Science and Systems, July 2023.
</content>
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</thread>


Metareview:
metareview:
The paper introduces the Planning Neural Operator (PNO), a model that predicts value functions for motion planning by learning an operator from cost function space to value function space, demonstrating zero-shot super-resolution, accurate predictions, and significant efficiency improvements in motion planning across 2D, 3D, and robotic manipulator tasks.

The reviewers highlighted several strengths of the paper, including (1) its novelty, (2) applicability across multiple tasks, (3) advantages over existing baselines (VIN, NTFields, and P-NTFields), and (4) its clear and well-structured presentation. However, they also raised concerns about the simplicity of the experimental setups and questioned the method's scalability to larger-scale, higher-dimensional spaces and real-world scenarios.

During the Author-Reviewer Discussion phase, extensive discussions occurred between the authors and Reviewer 6Xqc. The authors provided additional experimental results and clarifications that clearly strengthened the paper. While incorporating more challenging experiments, particularly in real-world settings, would further enhance the paper's impact, the AC acknowledges the theoretical contribution and believes it represents a valuable addition to the ICLR community.

The AC recommends acceptance but encourages the authors to carefully address both pre- and post-rebuttal comments to address any remaining concerns in future revisions.