Generalizable Motion Planning via Operator Learning

Thank you very much for your constructive and detailed review. Your suggestions have been extremely helpful for improving the quality of the paper. We also appreciate that you found the paper to be an enjoyable read, despite its mathematical details.

1. Clarification of practical applicability of the assumptions: Thank you for the question on clarification of the assumptions.

   • For Assumption 1, we require the existence of a proper policy. In practice, this assumption is satisfied when there is a feasible path from every start position to the goal. This assumption excludes trivial planning problems that may have no solution.
   • For Assumption 2, we require equicontinuity and uniform positivity for the space of possible cost functions. In practice, this assumption can be satisfied by creating smooth boundaries around each obstacle and associating them with strictly positive motion cost.

   We added sentences after both of these assumptions to clarify the practical meaning of the assumptions.

2. Scalability and more challenging environments: We thank the reviewer for their suggestion to utilize PNO in more complex environments.

   • We emphasize that by planning in continuous function space, our method is able to scale to large $2D$ and $3D$ environments. For example, there has been very little work that considers neural motion planning on environments of the size $1024 \times 1024$ which consists of $1$ million+ points. However, as the reviewer points out, this task is not necessarily the most complex because it assumes a point robot. Thus, to highlight the ability of our approach to handle scenarios with more complex robot configurations, we introduced a $4$-DOF manipulation task within the short rebuttal period. We show that our PNO architecture is able to generate viable value functions for manipulator motion planning in Figure 8 and highlight to examples of end-effector trajectories in Figure 4. We believe that extending the PNO to $4$-DOF manipulators in a short time shows the strong promise of the PNO design to handle high DOF robot systems. We acknowledge that additional improvements are needed in our architecture to eliminate the discretizaiton of the input cost function and enable scaling to high DOF configuration spaces. We think that the $4$-DOF experiments are encouraging for extensions to $6$+ DOF manipulators.

   • We leave implementation on real-world robots as future work, given that in this paper we focus on the learning architecture design. Particularly, this work introduces new connections between the neural operator approach and motion planning problems. Thus, we present an analysis both theoretically and pragmatically exploring the generalization and super-resolution capabilities of the general learning algorithm across a variety of different tasks (2D planning in real city maps, 3D iGibson navigation, 4D robot manipulator planning, and deployment as a neural heuristic in classical motion planning algorithms).

3. Ablation study: We intended for the MovingAI city maps to highlight the architecture advantages of the triangle inequality preservation and obstacle encoding in our model. To clarify this, we have made multiple additions to the Moving AI city maps section and highlighted the effects of the PNO in Table 2. In Table 2, we see that the improved performance of obstacle encoding when comparing FNO (without obstacle encoding) to PNO. Further, we highlight the generalization improvement of the triangle inequality output layer in the same Table when comparing PNO and DAFNO (the latter does not include the triangle inequality). We thank the reviewer for helping us better showcase the advantages of our approach with this suggestion.

4. Table 1: We identified the issue with the results on the Grid-World dataset. The poor performance of our method is attributable to the FMM solver, which computes value functions in continuous space. States that are laterally surrounded by obstacles in the x and y directions become effectively unreachable because they can only be accessed diagonally while grazing the obstacles—a scenario that the FMM solver does not permit as it is considered a collision. Thus, the PNO model, which is trained using the FMM solver outputs as labels is unable to correctly generate paths when the start and goal positions require a diagonal movement between two touching obstacles. These types of maps are most prevalent in the $16\times 16$ dataset and actually are one of the major issues stopping PNO from reaching a higher success rate in all three map sizes. If one trained our PNO model with Dijkstra generated labels that account for these diagonal movements, the issue would be alleviated. However, to stay consistent, we did not modify the FMM solver nor the maps to ensure a fair comparison with the other benchmarks.

Thank you for your constructive and detailed review. Your suggestions have been extremely helpful for improving the quality of the paper.

1. Pure PINN loss: A key advantage of our theoretical formulation is that it encompasses a wide class of operator networks including physics informed neural operators (PINO, [1]). As such, one can certainly employ a pure PINN loss to train a network without data labels under our framework. This would enable discretization-invariant training without supervised labels but would most likely sacrifice the generalization as PINNs typically require retraining for every new environment. Moreover, it is also possible to use a hybrid approach. Namely, instead of a pure PINN loss, one can utilize a combination of the supervised approach with a PINN loss correction term, which has been explored for PDEs in [1]. This would of course require labeled data, but perhaps only for a few environments which is then achievable to compute - In fact, one could also perhaps use neural network generated labels from P-NTFields/NT-Fields to create these labels paying a large upfront cost offline in exchange for computationally fast deployment in a variety of environments online.

2. Testing on general geometries: We very much appreciate this suggestion as it highlights the power of connecting robot motion planning with operator learning in that we can take advantage of the existing literature that has been successful in the SciML community. However, one challenge arises in that our methodology encodes specific properties (i.e., obstacles and triangle inequality) in the value function. It is not obvious how those properties would be enforced if the input requires a latent space representation via an autoencoder along the lines of references [1] mentioned by the reviewer. This is an exciting future direction to consider.

   We can, however, work directly on general geometries by circumscribing the geometry with a rectangle and then using $\tilde{\chi}_{PNO}$ as in the PNO framework to treat the points not in the geometry as obstacles. With this approach, one can then apply PNO on any desired geometry.

3. Presentation: We made the following revisions to improve the presentation.

• We added contour lines to the value functions in Figure 3.
• Additionally, we added sentences throughout Section 3 to describe the components of our architecture mathematically. We thank the reviewer for helping us improve the clarity and therefore the usefulness of our approach.

4. Baselines: As addressed above, we added comparisons with P-NTFields (See above and Figure 3 of the revised manuscript).

[1] Z. Li, H. Zheng, N. Kovachki, D. Jin, H. Chen, B. Liu, K. Azizzadenesheli, and A. Anandkumar. Physics-informed neural operator for learning partial differential equations. ACM/IMS J. Data Sci., 1(3), May 2024

Thank you for your response and furthermore thank you for your entire discussion throughout this rebuttal period.

We have conducted additional experiments as requested by the reviewer showcasing the training of a PNO with a hybrid PINN + relative $L_2$ error loss on the 2D dataset given the short response window. The loss function is explicitly given by Avg $L_2$ relative error + $\lambda \left(\sum_{x \in \mathcal{S}} (\lVert \nabla u(\mathbf{x})\rVert - c(\mathbf{x}))^2\right)^{\frac{1}{2}}$ with $\lambda=0.05$ as a scaling hyperparameter. Although the PDF is not currently, editable, we promise to include such experiments in the final revision of the manuscript.

We first present the following table for the 2D dataset of which we ensure is extensively detailed for the reviewer as both an ablation study and verification of our approach.

Given the results, it is clear the PINN loss does improve the performance of PNO and we very much thank the reviewer for their suggestion. We also see it significantly improves the generalization performance when applied to super-resolution of the PNO.

Thank you very much for your constructive and detailed review. We appreciate your suggestions for improving the paper! As aforementioned, we added a planning experiment on a $4$-DOF robot manipulation task in Figure 4 and Figure 8. We believe that extending the PNO to 4-DOF manipulators in the short rebuttal period shows the strong promise of the PNO design to apply to high DOF motion planning problems. In the future, we plan to extend the PNO architecture to real-world $6$ and $8$ DOF manipulators. We acknowledge that the current cost function representation needs to be generalized to achieve scalability in higher DOF problems. One approach would be to invoke an autoencoded representation of the geometry of which neural operator approaches exist as in [1]. Hence, it may be possible to preserve the discretization invariance property of PNO in this setting but it is not obvious how to ensure the satisfaction of the triangle inequality and the hard obstacle constraints in the latent space. We leave this as an exciting direction for future work.

[1] J. H. Seidman, G. Kissas, G. J. Pappas, and P. Perdikaris. Variational autoencoding neural operators. In International Conference on Machine Learning (ICML), 2023.