Generalizable Motion Planning via Operator Learning
Authors: Sharath Matada
Luke Bhan
Yuanyuan Shi
Nikolay Atanasov
Venue: ICLR 2025 Poster (2025)
Keywords: Motion planning
operator learning
value function approximation
Eikonal PDE
Primary area: applications to robotics, autonomy, planning

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$^\ast$).

Published as a conference paper at ICLR 2025
GENERALIZABLE MOTION PLANNING VIA OPERATOR LEARNING
Sharath Matada; Luke Bhan, Yuanyuan Shi, Nikolay Atanasov University of California, San Diego (smatada, Ibhan, yyshi, natanasov @ucsd.
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 approxima- tion 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 finite number of samples at coarse resolution, inherits the zero-shot super-resolution property of neural operators. We demonstrate accurate value function approximation at 16x the training resolution on the Moving lab's s 2D city dataset, compare with state-of-the- neural value function predictors on 3D scenes from the iGibson building dataset and show- case optimal planning with 4-DOF robotic manipulators. Lastly, we investigate employing the value function output of PNO as a heuristic function accelerate motion planning. We show theoretically that the PNO heuristic is e-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 MovingAl lab 2D city dataset, compared to classical planning methods (A*, RRT'). See project code https:: //g::////ttti www.bbt..huuttgggggggggggg. com www.istentialRobooeooo iPNNNO.
INTRODUCTION
Classical tools for motion planning in complex environments face performance degradation as the scale the environment increases. To alleviate this issue, modern approaches introduce neural network models enabling computational efficacy (Lehnert et al., 2024). Recent connections have been made between motion planning and scientific machine learning in the context of solving partial differential equations (PDEs). Specifically, the motion planning problem be formulated from an optimal control perspective leading to an Eikonal PDE whose solution yields the optimal value function. Recent work (Ni & Qureshi, 2023a) has explored solving the Eikonal equation via physics informed neural networks (PINNs) without the requirement of labeled data. However, as with classical PINNs, these approaches require retraining for each environment and, therefore, are computationally intractable for recomputing the solution quickly in dynamic environments governing the real-world.
In this work, we reformulate the solution the Eikonal equation as an operator learning problem between function spaces. This new perspective enables the training of single neural operator which maps an entire space of cost functions, with each cost function representing a separate environment, to corresponding space of value functions. We design a new neural operator architecture, called Planning Neural Operator (PNO), that learns the solution operator of the Eikonal PDE and thus the value function. Due to our architecture design, PNO generalizes to new environments with different obstacle geometries without retraining. Furthermore, we capitalize on the resolution invariance property of neural operators, enabling training with coarse resolution data and deployment of the learned neural operator on test maps with 16x the training data resolution as illustrated in Fig. 1. In summary, our contributions given follows:
(a)
We formulate the motion planning problem as an optimal control problem whose value function satisfies the solution the Eikonal PDE. We then prove arbitrarily tight accuracy, the existence of a neural operator approximation to the Eikonal PDE solution operator.
"quual contribution

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Input Mask
Ground truth (FMM)
FNO SDF
Planning neural operator (PNO)
tht
512
1024
Figure 1: Example of super-resolution capabilities of PNO for motion planning map of NYC. The operator is trained on a dataset of resolution 64 x 64 and the examples shown here (resolutions 256 x 256, 512 x 512, and 1024 1024) were not seen during training. See Sec. 4 for details.
(b) We design a Planning Neural Operator (PNO) to approximate the solution operator of the Eikonal PDE and enable generalizable motion planning. By design, the architecture is (i) resolution invariant, (ii) encodes obstacle geometries into the learned weight space to enable generalization across different environments, and (iii) introduces projection layer satisfying the triangle inequality to enable generalization across goal positions.
(c) We develop four experiments to highlight PNO's advantages. These include zero-shot super resolution, optimal path planning on 3D iGibson building environments, 4-degree of freedom (DOF) manipulators, and deployment heuristic function in A reducing the expanded nodes by 33% while maintaining near-optimal paths.
Related work. This work builds on ideas from two fields PDEs and motion planning. Thus, we briefly review popular motion planning approaches while identifying connections between motion planning and PDEs and conclude with a brief overview of operator learning.
Motion planning. Motion planning techniques can classically be split into two categories sample- based algorithms and search-base basedd algorithms. Sampling samples in configuration space (C-space) to construct graph structures whose edges connect to form collision-free paths. The two most widely known algorithms are probabilistic roadmaps (PRM) (Kavraki et al., 1996) and rapidly-exploring random trees (RRT) Valle, 2006; Karaman & Frazzoli, 2011) Since the development of these methods, there have been significant extensions using ideas such as biased sampling with heuristic functions (Gammell et al., 2015; 2014), parallelization (Sundaralingam et al., 2023), domain decomposition (Chamzas et al., 2021), and potential fields (Qureshi & Ayaz, 2015). We refer readers Orthey al. (2024) for review.
motion planners generate random
In contrast with sampling-bant planners, search-based planners trade speed in favor of obtaining optimal paths. The most well known approaches are Dijkstra's 's algorithm (Dijkstra, 1959) and A* (Hart et al., 1968; Likhachev et al., 2003), which prioritizes the search order using a heuristic function. More recent search-based planners include Cohen al. (2010), Liu al. (2018) which plan using motion primitives. Lastly, we briefly mention planners based on Eikonal solvers such as Sethian (1996), Janson et al. (2015), Valero-Gomez et al. (2013), Pêtrès et al. (2005). These approaches, instead of planning by exploring nodes in configuration space, aim to solve the Eikonal PDE yielding an optimal function for the resulting environment and then perform corresponding gradient descent.
Despite the success of classical planning methods, they suffer from computational limitations when scaling to higher dimensions. Thus, with the recent success of deep learning, neural motion planners (NMPs) have exploded in popularity for reducing the computational burden of motion planning. For example, many works such as value iteration networks (VIN) (Tamar et al., 2016), motion planning networks (MPN) (Qureshi al., 2021) well et al. (2019),Cm Chaplot et al. (2021) and Fishman et al. (2022) attempt to plan completely from neural networks designs. Moreover, other approaches instead augment classical algorithms. For example,
recent approaches such Zeng
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Chen et al. (2020), Zhang et al. (2022) speedup the collision checker in RRT and Ichter et al. (2019), Wang et al. (2020) learn sampling biases. See McMahon et al. (2022) for a comprehensive review. Further, some approaches use gradient information directly from either Neural Distance Fields (Ortiz et al., 2022; Liu et al., 2022) or Neural Radiance Fields (NeRF) ( (Adamkiewicz et al., 2022). Lastly, mention three approaches that are most close to ours. The first, NTFields/P-NTFiille Qureshi, 2023a;b) learns PINN for motion planning based on the Eikonal equation, but requires retraining for new environments. Meanwhile Li al. (2022) learns implicit functions for planning but has no guarantees of obstacle avoidance and pays large startup costs for data generation.
(Ni &
Operator learning. The goal of operator learning is the design of models for approximating mappings across function spaces. Neural operators were first introduced in a seminal paper by Chen & Chen (1995) that designed the first architecture and a corresponding universal operator approximation theorem for approximating PDE solutions. This work was recently rediscovered and extended using modern neural network models under the DeepONet and FNO frameworks (Lu et al., 2021; Deng et al., 2022; Lanthaler et al., 2022; Li et al., 2021; Kovachki et al., 2023; 2021). Since, there have been a series expansive papers addressing numerous challenges from non-uniform geometries (Liu et al., 2023; Li et al., 2023a;b; Fang et al., 2024) to architectural limitations (Seidman et al., 2022; You et al., 2022a; Gupta et al., 2021; Hao et al., 2023; Furuya et al., 2023) of the original works. Furthermore, neural operators have been employed in eclectic range of applications spanning, but not limit to, weather forecasting (Kurth et al., 2023), material modeling (You al., 2022b), scismic wave propagation (Yang et al., 2021), and control of PDEs (Bhan et al., 2023). Lastly, from a theoretical perspective, Lanthaler et al. (2023) unified the neural operator framework, introducing an abstract formulation of the operator learning problem and a corresponding universal approximation theorem (see Theorem 2) that encompasses wide array of architectures, including Fourier neural operator (FNO) and DeepONet.
EIKONAL PDE FORMULATION OF THE MOTION PLANNING PROBLEM
2.1 MOTION PLANNING AS OPTIMAL CONTROL. PROBLEM
Consider a continuous- dynamical system with state x(t) E X C R", compact state space X' control input u(t) 2 c 12 compact control space U, and dynamics:
ax( [((((),,
(1)
Given an initial condition (0) = Co. we aim to design a control policy X - U that maintains the system state (t) in safe set S c X' for all t and drives it to goal set 9 C S. We formulate this problem through an affnite-horizon first-exit optimal control problem (OCP) as Bertsekas (2017)
min c-(x(T)) I." ((((),=(((()))d):
(2a)
s.t. T= inf Rgo (+1) E g), ((()),
(2b)
=00)= = mn. sx1) E S V2
(2c)
where is the first-exit time, the smallest time at which the system reaches the goal region G. c: X x U + R is the stage cost function that measures the quality of the system trajectory, and C X + IR is terminal cost function evaluated when the system reaches the goal region. To ensure the well woll-posednees of (2), we enforee the following standard assumption (Liberzon, 2012). In practice, this assumption is satisfied when there is feasible path from every start position to the goal.
Assumption 1. There exists a policy π : X U such that, for any initial condition (0) E S, the trajectories of the closed-loop system with utt) (xx(t) in (1), reach the goal region G by finite first-exit time T < oc.
As standard, the optimal value function V : X R is the minimum cost attained in (2) given by
V(x) == min (c + L' ((x(t),((<(()))))
(3)
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subject to the constraints in (2). A sufficient condition for optimality is that the function V satisfies the Lamilton-Jacobl Jalll--elma equation:
0 = min (V(x)' f(z, u) + c(x, u)}
Væ E S G,
(4a)
wEM/
V(x) (z),
Va g,
(4b)
where VV denotes the gradient of V. As in (4a), (4b), to simplify notation, we omit the time argument for æ(1) and ((x(()) in the remainder of the paper.
2.2 EIKONAL EQUATION FOR OPTIMAL MOTION PLANNING
In this work, we capitalize on the structure of common motion planning problems to simplify the HJB equation. In practice, one commonly splits the OCP described above into two subproblems: i) a motion planning problem, which determines reference path from the initial state (00) to the goal region & without considering the dynamics constraint in (2c), and ii) control problem, which determines a control policy for the system to track the planned reference path. We focus on the planning problem, which simplifies the system dynamics to be fully actuated, (+()) u(().
To ensure feasible paths, it is common to limit the magnitude of the control input u(t). Thus, we constrain the input the input magnitude is already constrained by U it is not necessary to penalize it in the stage cost. Hence, we let c(ær, u) = c(x) be control-independent cost, and let e, (x) = c(x) for presentation simplicity. Thus, the optimal control problem in (2) reduces to the optimal motion planning problem:
admissible control set U {u |||| = =1} of unit vectors. When
min dx((T)) + / c(x(t))dt,
(5a)
s.t. r in ft Rzu xxt) E g}.
(5b)
(=t) = (xx()) z00) Fo.
(5c)
(+t) E S, [[(((( = 1, 41 0,
(5d)
and the associated HJB equation in (4) simplifies to:
min (VV(z) u + c(x))
Va S \ 9,
(6a)
uEll
V(x) c(x))
Va E Eg
(6b)
Note that the minimization in (6a) is now linear in u with the constraint ||u|| = 1. The minimizer is readily computable in closed-form u = [VV(()]] yielding PDE in the Eikonal class:
Va()))
[V]())] = c(x: (ax) = c(x),
Va E S 9,
(7a)
Vo E G.
(7b)
2.3 PROPERTIES AND UNIVERSAL APPROXIMATION OF THE EIKONAL SOLUTION OPERATOR
The solution to the Eikonal PDE in (7) be viewed infinite dimensionll nonlinear operator that maps cost functions c(x) into corresponding value function solutions V(x). However, given an arbitrary cost function c, there no known analytical representation for the operator V (c).. Thus, in this work, we ask if it is possible to learn an approximation to the operator V and, if so, how to design a resolution invariant neural network to approximate T/? To start, we answer the first question affirmatively by theoretically proving the existence of such a neural operator approximation.
To theoretically show the existence of Ú we begin with brief review of neural operators. A neural operator U is neural network architecture consisting of a lifting neural network, kernel approx- imation neural network, and a projection neural network (See Appendix for full formalization). Under such an architecture several universal approximation theorems exist of which we focus (Lanthaler et al., 2023, Theorem 2.1) (restated as Theorem in Appendix B. due to its abstract formulation that encompasses variety of architectures.
Under the universal approximation theorem, there are two challenges to establishing the existence of a neural operator & for approximating the Eikonal PDE solution. First, we require that the domain is compact set of continuous functions and second we require continuity of the operator. For the first
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condition, since the solution to the Eikonal PDE is not continuous, aim to learn the continuous and unique viscosity solution (see Appendix C). Let Fe be the function space of costs X' + R. namely cl+) E Fe. Likewise, let F,, be the function space of value functions X' R. that is V(-) F... To ensure well-posed solutions of Problem (7), we require the following assumption.
Assumption 2 (Continuity and uniform boundedness of cost function space). The cost function space Fe is uniformly equicontinuoos. Further, there exists a constans 0 > 0 such that
inf cxx) > 1/0, sup c(x² < Ve E Fe. =EX
(8)
=EX
The set Fe needs be equicontinuous ensure the continuity of the operator V. The uniform positivity of the cost e is required to ensure the existence and uniqueness of viscosity solutions to the Eikonal PDE. In practice, this assumption can be satisfied by creating smooth boundaries around each obstacle associated with strictly positive motion cost. Lastly, by the continuity of cost functions in Assumption and the Arzelá Ascoli theorem (Folland, 1999, Theorem 4.43), Fo is compact. From these assumptions, a viscosity solution V F. F, to the Eikonal PDE always (7) exists (See Appendix C). We are now ready to present the existence of a neural operator approximation to V. Theorem 1. (See Appendix A. for proof) Let Assumptions 1, 2 hold and consider V : F + Fu as the solution to (7), Then, for any >> O, there exists a neural operator ip : F. - Fo such that
sup oEF
(c) (r() Se
While Theorem guarantees the existence of a neural operator for approximating the Eikonal PDE, does not include the construction of the neural operator V. This the focus of the next section.
DESIGNING NEURAL OPERATORS FOR EIKONAL PDES
We introduce a new neural operator architecture, which we call Planning Neural Operator (PNO), to approximate the solution operator (c) V of the Eikonal PDE in (7). We ensure that our architecture achieves three goals: (i) invariance to resolution differences between train and test maps, (ii) generalization across environment geometries, and (iii) generalization across goal positions.
To achieve property (i), we employ the resolution-invarian Fourier neural operator (FNO) architecture (Li et al., 2021) with two key extensions. First, to enable generalization across environments, we hard encode the obstacle geometries into the operator structure. Second, to generalize across goals, we design the output layer of the FNO model to ensure that the predicted value function satisfies the triangle inequality. Particularly, when the goal set is a singleton G = {g]. let V(z, g) denote the solution to (7) with goal g. Then, by the principle of optimality (Bellman, 1957), V satisfies:
V(æ,g) V( y) (V.,), Va,
(10)
Thus, we seek a neural operator V(x, g) = (ύ (zx) that encodes this property of the value function. Next, review the structure of the FNO model, which ensures property (i). Then, present our PNO architecture that extends the FNO model to enable properties (ii) and (ii)..
3.1 REVIEW OF FNO AND DAFNO
An FNO model consist of three components: (1) a lifting neural network that maps to high dimensional space, (2) a series of Fourier layers, and (3) a projection neural network to the target resolution. It can be viewed as mapping V = Q o LM o ...o L1 a R, where R(c(x), x) is the lifting network, Ly for m 1, 1,.... ... .M are the hidden Fourier layers, and Q is the projection network (See Appendix B.1 for details). The Fourier layers consist of applying a Fast Fourier transform (FFT) to the input, multiplying that transform by learnable weight matrix and then transforming back via an Inverse FFT. Mathematically, this is formalized as L. (x) ;= - -(We.. (x)) where F - are the Fourier and inverse Fourier transforms and We,,, is learnable, complex valued weight matrix. The key idea is that, by learning in frequency space, the operator is resolution invariant as can project any desired resolution to and from a pre-specified number of Fourier modes.
However, to perform Fourier transform. FNO needs rectangular domain and thus cannot be applied to non-uniform geometries. Recently, a new model, Domain Agnostic FNO (DAFNO) (Liu et al., 2023) addressed this issue. DAFNO circumscribes the non-uniform geometry of the solution
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Solves value function
Inductive hlas on function
Solves SDT Eikonal PDE
"(N -rvo -(())
DAFNO Layers
obstacle
encodes
fs(d(x) dkgg)
(Wg F(x))
Despnorm
FND Layers
Goal position
g
Ensures triangle
inequality satisifed
Signed distance fuaction ds
Rinary oceapancy map
Value function VCK, g)
Figure 2: PNO network architecture. The input to a PNO is 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.
operator into rectangle and then encodes, in the Fourier layer, where the true solution domain active. To achieve such an encoding. DAFNO explicitly introduces an indicator matrix of the active solution geometry X into the Fourier layer multiplication. Explicitly, this is expressed by the following modified layer LA,DO == & (Wo.. X x-x(())) where X is the aforementioned indicator function (Details on the DAFNO architecture are in Appendix B.2). In PNO, we take inspiration from DAFNO by explicitly encoding the obstacles using similar indicator function.
3.2
PNO ARCHITECTURE: GENERALIZING ACROSS ENVIRONMENTS
The PNO model maps cost function to the corresponding value function of the motion planning problem by explicitly designing the layers R, Li..... L., Q to achieve goals (ii) and (iii). However, before discussing the layer design, we begin by formalizing the input to a PNO, We consider minimum-time motion planning problem by choosing the cost function as
x: E.
c(x)
(1I)
x E X
where penalize the unsafe set with infinite cost. In practice, to avoid numerical instability, use 1/c(z), or a binary occupancy map as the input function. We then employ a neural network for our lifting layer in FNO, that is (1xo(1/((z)) are learnable weight tensors. To achieve property (ii), we ensure that PNO explicitly captures the obstacle configuration of the environment. That is, hard encode the obstacle locations in the Fourier layer of our architecture. To do so, taking inspiration from DAFNO, we modify the Fourier layer by multiplying the learnable weight matrix with the following smoothed indicator function
Wo,, (1/e(x)) Bo, where Wo, and Bø
rrno a=) tanh
0.5) 0.5,
(12)
where B is smoothing hyperparameter
inf x
yll. if E
ds(x)
wEas
(13)
inf x ull. if o S,
yEaS
where as is the boundary of S. Thus, the inner layers of PNO are given by Im, f (W- XPPN F(x), m = I,...,,. M. Multiplication by XPNO achieves two goals. First, ensures that the indicator approximation (12) continuous and thus retains the guarantees of Theorem 1. Second, for each environment, PNO ignores the unsafe space in the weight matrix multiplication, which greatly improves generalization to new obstacle configurations.
In large motion planning problems, computing the SDF in the smoothing function (12) can be computationally challenging. However, note that SDF ds (x)) is itself solution to an Fikonal PDE:
[V) =
E X, ds(x) O, x E as.
(14)
Thus, under Theorem 1. there exists neural operator approximation to (14). As such, introduce a second neural operator, in the form of an FNO, trained independently, that maps from a binary occupancy map the corresponding SDF, namely 1/c(zz) + ds()). Thus, the occupancy function

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1/c(z:) and the SDF (z) are taken as inputs to generate the smoothed indicator in (12). This is then used to modulate the kernel in the Fourier layers of PNO. Since we use an FNO to generate the SDF function, our entire architecture maintains property (i), namely resolution invariance. Lastly, inspired by PINNS for motion planning Ni & Qureshi (2023a), we introduce a PINN loss:
Loss(V, V. P) == IV - VIL + ( (IV )II - c((x)) 1/2
(15)
where ç controls the weighting of the PINN component. Note, i > 1, the PINN loss dominates resulting in any solution that satisfies the gradient (c.g. Euclidean norm) and thus must be tuned.
3.3 PNO ARCHITECTURE: GENERALIZING ACROSS GOAL POSITIONS
Given the lifting layer and the modified Fourier layers discussed above, we move design the projection layer of PNO. The projection layer contains two inputs, namely the output from the last hidden layer and the goal location g. This goal location is equivalent passing the boundary condition (7b) for the solution operator V into our network. To enable goal generalization (iii), we leverage the fact that the optimal value function must satisfy the triangle inequality. Thus, we design our output with an inductively biased Deepnorm layer Q(---) (Pitis et al., 2020) given by:
QpNolo, x, = fng (ó(x)) - o(g)),
(16)
where o = LM.PNO o... LI, RpNo. and thus ((x) <(g) is the subtraction of the feature vectors between æ and the goal g. feg consists of regular neural network layers with non-negative activation function (e.g., RcLU) and positive weight matrix W+ This ensures that our value function satisfies the triangle inequality with respect to g (Pitis et al., 2020) and enables generalization across different goal positions (Section 4). Lastly, summarize the PNO architecture in Fig. 2, starting from a binary occupancy input 1 //((z)) and returning a value function as the output.
LEARNING MOTION PLANNING VALUE FUNCTIONS
To validate the efficacy of our PNO architecture, we design four experiments, First, we test our method on grid-world environments comparing PNO against learning- based motion planners. However, given the simplicity of the small grid world dataset, the environments lack the complexity of real-worle tasks. Thus, we introduce three real-world experiments. The first experiment highlights the super-resolution property of PNO by training on small synthetic maps and evaluating on large real-world maps from the Moving AI 2D city dataset (Sturtevant, 2012). In remaining experiments, we showcase the scalability of our approach in 3D iGibson environments (Shen et al., 2021) and 4DOF manipulators.
We compare our method against the FMM (Sethian, 1996) which solves the Eikonal PDE numerically, state-of-the-art neural motion planners: VIN (Tamar et al., 2016), NTFields (Ni & Qureshi, 2023a), P-NTFields Ni & Qureshi (2023b), IEF2D (Li et al., 2022), and two operator learning architectures: FNO (Li et al., 2021), DAFNO (Liu et al., 2023) (Baselines details in Appendix D).
Grid-world environments. In the first ex- periment, we consider a small Grid-Worle dataset as in Tamar et al. (2016), consisting of 5k training maps and 1k testing maps. We compare with VIN and IEF2D. For planning, we perform gradient descent on the test-map value function predictions from VIN, IEF2D, and PNO. In Table 1, we present the computa- tion time needed for value function prediction and the success rate of reaching the goal. We
Table 1: Comparison of planning on learned value functions on the Grid World dataset at various sizes.
Avg. success
Avg. computation time (ms)
rate t
82
16a
28a
g2
162
28
VIN
99.6
99.3
96.7
3.9
22.7
82.1
IEF2D
99.7
98.3
97.0
13.3
14.8
20.4
PNO (ours)
99.9
97.3
99.3
2.6
4.7
6.4
can see that all baselines perform quite well, with PNO outperforming both methods the larger 28 x 28 sized mazes while achieving an improved computational time. In Appendix D, we give an example of the learned value function and corresponding map for the PNO. The superior performance of PÑO on the Grid World dataset is primarily due to the fact PNO satisfies the triangle inequality and guarantees obstacle avoidance where both VIN and IEF2D lack such properties.
Ablation study: Moving AI 2D city maps. To highlight the advantage of reformulating motion planning as operator learning problem in continuous function space, we demonstrate that our
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PNO architecture be trained with a coarse resolution and deployed at finer super- rupertreoouttion on the Moving AI 2D real-world city maps (see Fig. 1). For training, we use a synthetic map dataset generated at 64 x 64 resolution (See Appendix D.3). We then evaluate the super-resolution performance of our method in comparison with two operator learning architectures Table 2. These results constitute ablation study of the obstacle encoding and Deepnorm components of our PNO model (see Fig. 2). In particular, DAFNO and PNO outperform FNO on the synthetic and real-world datasets, highlighting the importance of explicitly encoding the obstacle geometry into the network structure. Furthermore, we see the effect of the Deepnorm layer as PNO generalizes better compared DAFNO, particularly in the unseen real-worke city dataset achieving a 50% reduction in L2 error. In the third column of Table 2, we see the computational speedup at scale of the PNO architecture. In particular, the SDF generation significantly slows DAFNO and we achieve almost 10x speedup over the fast marching method (FMM). Lastly, we provide an example of navigating New York City in Fig. I where we showcase that the FNO is able to learn the SDF accurately (relative L2 test error of 0.064) and that the corresponding value function is accurately computed at all three super-resolution scales.
Table 2: Average L2 value function error of FNO, DAFNO, and PNO versus FMM.
Avg. relative
Avg. computation
Map Size
64
256
512"
1024
64
256
512-
1024
64
256
512-
1024
Synthetic obstacle datasel
MovingAl real world city dataset
real-world
Synthetic real world real-world city
(100 maps, in-distribation)
(90 maps, -istirdiistribution)
(1000 maps, ms)
ENO (PNO w/o Doepoor
obstacle encoding)
0.1996
0.5771
0.6214
0.6405
0.7188
0.7519
0.7692
1.62
1.64
1.66
2.28
DAFNO (PNO wlo
Deepnorm layer)
0.0985
0.3868
0.4060
0.4120
0.4090
0.4259
0.4315
3.10
4.97
11.44
49.82
PNO (ours)
0.1136
0.1197
0.1190
0.1194
0.1748
0,1885
0.2034
5.57
5.31
6.33
8.31
PNO w/ PINN loss (ours)
0.0698
0.0865
0.0869
0.0872
0.1675
0.1761
0.1842
5.57
5.31
6.33
8.31
[numerical solver))
Bolton environment NTFields
FMM
P-NTFields
PNO (ours)
Samuel environment NTFields
FMM
P-NTFields
PNO (ours)
Average relative
computation time (100 inst,
Lg Brllio En un-lronment
PRO (ound)
6.0711
$1.50
NY
0.213
$6,07
P.NTFaak
0.3008
39.56
FMM
05
Sammel Emviroment
PNO
0,2413
NYTieds
0.119
33,00
PA PKTThkks
Figure 3: (left) Comparison of 3D value function approximations for two iGibson environments (best Ta PNO, sliced at = = 0). (right) On top, show example paths in the Bolton environment. On the bottom table, we present a aggregate quantitative comparison. See Appendix D ..4 for more examples. 3D planning in iGibson environments. In this experiment, we consider planning in 3D real-world iGibson enviromments (details in Appendix D.4). For comparison, we train NTFields and P-NTeields models the Bolton and Samuel environments which takes approximately 3 hours to train per environment (NVIDIA A100 GPU). To validate the generalization of our PNO, we ensure that our training dataset contains instances of the Samuel environment, but does contain instances of the Bolton environment with different start goal positions than during testing. Generating our dataset takes approximately 1 hour and the model training takes 4 hours (NVIDIA A100 GPU).
Fig. shows slices of the predicted value functions, where all four methods capture the general structure. On the top right, showcase example of planning. NTFields, due to local minima, is unable to guarantee valid paths whereas P-NTiields PNO and FMM reach the goal successfully. In table of Fig. 3, we that, quantitatively, method performs well predicting the exact value
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Goal
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).
function of FMM. Furthermore, P-NTFields does not accurately capture the exact magnitude of the FMM value function, but generates smooth paths due to effectively capturing the overall shape and thus the gradient direction. Finally, all the neural network approaches perform faster then the numerical solver achieving speedups of 2x (NTFields) and 3 (PNO, P-NTFields) over FMM.
Planning with 4 DOF Manipulator. We further highlight the efficacy of the PNO design by plamning with 4-DOF robotic manipulator. In particular, the PNO architecture learns mapping from 4 dimensional C-space representation of the binary occupancy grid to the corresponding value function. We learn grid of size 17 generating 40 maps with 10 randomized goal positions for training and 10 maps with 10 randomized goal positions for testing. The PNO achieves a 0.027 L2 relative training error and a 0.041 L2 relative testing error (Details in Appendix D.5.) We show examples of corresponding paths computed with classical gradient descent in Fig. 4. Furthermore, we showcase slices of the value function in Fig. 10 Appendix D.5.
5 LEARNED OPERATOR VALUE FUNCTIONS AS NEURAL HEURISTICS
Although one can plan on the learned value functions directly, they may exhibit local minima due to approximation error. To address this challenge, we capitalize on the rich set of classical planning algorithms by employing our learned value function as neural heuristic. Typically, motion planners such as A use the Euclidean norm heuristic as it is both fast to compute and guarantees an optimal path. However, the Euclidean norm ignores the geometry of the obstacles and thus regions surrounding obstacles can require extensive exploration before optimal path is found. Accordingly, we propose using the learned value functions from PNO as an alternative heuristic to the Euclidean norm. We prove that our heuristic is consistent and showcase the advantage numerically, achieving a 33% decrease in the number of explored nodes compared to the Euclidean norm.
We begin by showing that PNO value function is e-consistent heuristic, which is sufficient property for the A algorithm to compute an e-optimal path (Likhachev et al., 2003).
Definition Let V(x. g) be the value function with goal position g. A heuristic function h((a:) X' + R is said to admissible if h(x)) < V( g) for any E X. It is said to be consistent if h(xx) < V(x, y) + h(y). Furthermore, for any E >> 1, a heuristic is e-consistent if (f()) S <V y) + h(y). Lemma (e-consistency of neural heuristic). Let Assumption hold and let F. be the cost function space satisfying Assumption 2. Let eNO be the neural operator approximation error as in (9). Then, V (w), generated from from
the neural operator is an e-consistent heuristic with:
max (u
1 2eNo Cocooo/(eee ((x
(17)
Further, if one is interested a neural operator satisfying comsisicee for > 1, then the neural operator must have error worse than:
EO <
min (e.yESleyyyl
(e 11)V(x,))))
(18)
In continuous space, it is likely that e + 0o in Lemma 1. However, motion planning algorithms, such A", discretize the space yielding < co. Further, in Lemma 1, it is impossible to identify the

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Published as a conference paper at ICLR 2025
true approximation error CNO- Thus, we introduce a second, practical improvement. Our key idea to "erode" the obstacles (as in Fig. 5) with the intuition the neural operator applied to the eroded enviromment is more likely to be an inder-aprroxmmatinn to the true value function. This encourages admissibility which can improve the paths generated under the heuristic (Pearl, 1984). To conduct the erosion, remove the outermost layer from each obstacle (Serra, 1983) and repeat this operation several times depending on the environment size. Formally, this can be expressed as increasing the safe space as 3 S, where is the safe set after erosion and then computing the cost as in (11) with S. The following result guarantees that our eroded PNO heuristic does not harm the c-consistency. Lemma 2 (Eroded heuristic is "more" consistent). Let Assumptions 1, 2 hold, For any ENO > let (c), (6) be the learned solutions (7) satisfying Theorem with eN according to costs c, defined in (11) with S, S respectively. Let and be the e-consistent value functions generated by operators (c) and (c) with ENO error in Lemma Then,
To ensure we do not significantly underestimate the value function, we combine our heuristic with the Euclidean norm via h(x) max | gll. ([) while preserving e-consistency. We evaluate the impact of our PNO heuristic on the Moving AI lab city maps. For planning, we use A with our PNO estimated value function as the heuristic. In Table 3, we see that the PNO heuristic with erosion achieves near optimal paths while expanding 33% fewer nodes compared the Euclidean norm heuristic. Furthermore, note that without erosion, the PNO heuristic generates sub-optimal paths (Ablation study Appendix E). Lastly, in Fig. 5, we highlight single example of planning where the PNO heuristic yields an optimal path while expanding fewer nodes.
Table 3: Comparison of heuristics for A against classical RRT and RRT over 2D maps. The number of eroded layers is 12, 14, and 18 for 256 5122, and 1024 respectively.
subuppiimality
Avg. number
Arg. compulational
Avg. path length
estimate
nodes expanded
time (50 just., s)
Map sizee (2D)
256"
512
1024"
256
512"
1024"
256
512"
1024-
236"
512"
1024"
A
Euctidean norm
144,05
311.21
605.14
L,00
1,000
1,000
3310
14850
39965
0.186
0.868
3,606
A PNO (ours) w Erosion
144.05
311.72
608.54
1.000
1.000
1006
2024
9869
41394
0.136
0.684
2.759
A PNO (ours)
144.88
315.25
614.40
L015
8882
0.387
RRT
191.98
418.61
800.31
333
345
1322
0.014
0.645
0.066
RRT
177.29
403.56
785.83
1231
1.205
0.120
0.000
2.71
Euclidean norm
PNO value w/o erosion
PNO value w erosion
lP
PP
Figure 5: Path planning using various heuristics the Moving AI Shanghai city map. Each example on the left showcases the A 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.
CONCLUSIONS, LIMITATIONS, AND FUTURE WORK
We reformulated the motion planning problem as an Eikonal PDE and focused on learning its solution operator. To learn the operator, we developed the Planning Neural Operator (PNO) which is (i) resolution invariant, (ii) does not require retraining in new environments, and (iii) generalizes across goal positions. We evaluated our architecture on the 2D MovingAl city dataset and the 3D iGibson building dataset, showcasing super-resolution value function prediction while achieving speedups of 10x over a numerical PDE solver. Lastly, we proved that our value function predictions can be used e-consistent heuristics for motion planning algorithms, and demonstrated a 33% decrease in expanded nodes in the A algorithm compared to planning with a Euclidean norm hcuristic.
In the future, we aim to extend PNO with dynamic heuristic updates leveraging its fast planning inference. To address sub-optimal paths, one approach to invoke ARA" to repair inconsistent nodes where the second iteration on PNO's efficiency while ensuring optimality. Lastly, we considered cost functions with uniform step penalties. An exciting extension would be training PNO in environments with non-uniform costs.
1-consistent heuristic (e.g. Euclidean norm). This would capitalize
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ACKNOWLEDGMENTS
We greatly thank Jeff Calder's (jwealder@umn) (eneeruuunnedu) for his assistance in the proof of Theorem 1. We gratefully acknowledge support from Department of Energy (DOE) grant DE-SC0024388 Office of Naval Research ONR N00014-23-1-2355. National Science Foundation (NSF) 2120019 (CHASE-CI) and DOE DE-SC0025495. We also thank James Sorge for his assistance in various experiments. Sharath Matada thanks his family for supporting his research work throughout the Summer of 2024.
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Huaiqian You, Quinn Zhang, Colton J. Ross, Chung-Hao Lee, and Yue Yu. Learning deep implicit fourier neural operators (IFNOs) with applications heterogeneous material modeling. Computer Methods in Applied Mechanics and Engineering, 398: 15296, 2022b. ISSN 0045-7825. doi:
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15296. URL https: /ww.//wwwkkomom///owwo www.ssccciceeiee......... com/ /00///0045557 /ttp:///a...e/////2222000000
Wenyuan Zeng, Wenjie Luo, Simon Suo, Abbas Sadat, Bin Yang, Sergio Casas, and Raquel Urtasun. End-to-end interpretable neural motion planner. In 2019 IEEF/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 8652-8661, Los Alamitos, CA, USA, jun 2019. IEEE Computer Society. doi: N0.1100//VP..2211.00008 //doi. wwweecoopttersocccci org.1/........o 10t.119/////
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Ruipeng Zhang, Chenning Yu, Jingkai Chen, Chuchu Fan, and Sieun Gao. Learning-based mo- tion planning in dynamic environments using GNNs and temporal encoding. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (eds.), Advances in Neural Information Processing Systems, 2022. URL ht http::::s///pppeeveeeeee net hft./fooummmmmm wttee///oooommmmmmettttttttt
APPENDIX
A PROOFS OF LEMMAS AND THEOREMS
A.1 PROOF OF THROREM 1
Proof. We show & a continuous operator in L and then invoke Theorem noting that F, is compact by Assumptions 2 and the Arzelá Ascoli theorem. Since, Vi is the viscosity solution,
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satisfies the comparison principle (Katzourakis, 2014, Theorem 1). Thus, let c1, C2 Fe such that C1 I C2. Then, via the comparison principle, V (c1) I V (c2).. Furthermore, by the Archimedean property, find a real number À 0 such that λcι C2- A simple choice for such Á = x @he (x:). Then we have that (V())) 2 (c2) again by the comparison principle.
Finally, using substitution, Cauchy Schwartz, and then substitution again, we obtain
(c2) [ S (l(A
02 2-9
c1
(x)) [V(a))l
sup x
0,Balle22
ci cillz
(19)
where O, is the diameter of the value function space Fu. Thus v, Fo satisfy the requirements of Theorem completing the result.
A.2 PROOF OF LEMMA I
Proof. Let 4" (a) (2) be the optimal value function. The optimal value function is consistent, namely for any æ, y E X, we have h" h"(x)- h² (yy S V (* y). Now, for any ENO. there exists some (c) satisfying Theorem such that [c) - (c)] < ENO for any æ, y E X. Then, we have
h(z) hyy) S h" h((x) h" (y) + 2ENO
S V* y) + 2ENO.
(20)
To ensure,
that (20) is S (e 1) (a., y)
eV <V* v) for every 3, y. we can choose eNO
S
min { (a.ye\/aavu
Likewise, the smallest e achieved 6
max (a.yEX mzy)
2O
V'(x,))
A.3 PROOF OF LEMMA 2
(æ'(1,)) Proof. Let V' y) and V (æx y) be the optimal value functions for c, respectively. Then,
V* 2 V by definition of c, č. Let E(c), e(δ) be the minimum Eg satisfying Lemma 1, (20) for č respectivell Explicitly
V' (x, 8) + ZENO
(cc) = max (..yEX((
(21)
V=(x.V) v)
V' (x. y) + 2ENO
(()
(22)
(a.yEX{ m
V=(x,y) v)
Now, noting that V < V everywhere yields c(2) < c(c).
NEURAL OPERATORS
B. NONLOCAL NEURAL OPERATORS
Formally, we provide a review of the nonlocal neural operator (NNO) as in Lanthaler et al. (2023) under the architecture of the general neural operator first introduced in Kovachki et al. (2023). Such a framework is useful as it encompasses almost all neural operator architectures including the well known DeepONet Lu et al. (2021) and FNO Li et al. (2021). Let X' R" be bounded domain and define the following function spaces consisting of continuous functions Fe C (X';
F. C CO (X; Then, NNO is defined as mapping V Fo(X' R) + IR) which can be written in the compositional form v Q L., o... a L1 R consisting of a lifting layer R. hidden layers Lt, 1 1,.., I., and a projection layer Q. Given channel dimension de, the lifting layer R
given by
R -
c(x) R(c(x),)))
(23)
where F.(XR, is a Banach space for the hidden layers and R : R X + RR is a learnable neural network acting between finite dimensional Fuclidean spaces. For I = -,..., each hidden
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layer Ci is of the form
(Cx)) := a (Wio(z) + + (X))
(24)
where weights Wi E gd. and biases bj E Rd. are learnable parameters, IR + IR is smooth, infinitely differentiable activation function that acts component wise on inputs and 1C is the nonlocal operator given by
(KC))())= = = / Ki(a, (y)
(25)
where (x y) is the kernel containing learnable parameters given in various forms. For example, in the FNO architecture, Ki y) Ki( y). Ki(x) 1k Skun ee is a trigonometric polynomial (Fourier) approximation with knnx nodes and Pr, is a matrix of complex, learnable parameters Pt,: Cdexde, Note that (24) almost traditional feed-forward neural network except for the kernel term (25), that is nonlocal it depends on points over the entire domain rather then just æ. Lastly, the projection layer Q is defined as
@ F: + RR, where Q is a finite dimensional neural network from Rd x X - IR yiclding the final value of the operator (fe) (c E F.) at the point æ E X.
s(x) + ((s(x),))
(26)
Theorem 2 (Neural operator approximation theorem Lanthaler al. (2023, Theorem 2.1)). Let X' C Rn be a bounded domain with Lipschitz boundary and X its respective closure. Let y C R): + CO(2' R)) be a continuous operator, where C (2;; R) is the set of continuous functions X P-R. R. Then for any E and some compact set K. CO (2; (R;R), there exists a nonlocal neural operator V K.C (x; + (X: R) such that
sup (c) (c)) S. aEk
(27)
B.2 DOMAIN-AGNOSTIC FOURIER NEURAL OPERATORS
Domain-Agnostic Fourier Neural Operators (DAFNO), first introduced in Liu et al. (2023), augment the FNO with a mask such that FNO's 's can be applied on non rectangular geometries despite the requirement of Fourier transforms to operate on periodic domains. For the motion planning problem, as above, consider partitioning X' into two domains S for the safeset and X'S as the unsafe set. Furthermore, if X' is not box, we consider the smallest rectangle T that contains X' (see Liu et al. (2023, Fig. 1)) and add all the padded points in X''' to the unsafe set. Then, consider the following two characteristic functions that encode the geometry
x ESS
x(x) = o
(28a)
x x X
x(x) == (=n) 0.5) + 0.5,
(28b)
where 3 E IR is chosen hyper parameter parameter and ds is the sign distance function (SDF) given by
iiffuexxs | -x=10 yll
if e
(xx 1 inf || - yl
(29)
if $ Xs.
The idea is that x masks the geometry, setting the domain to 0 where obstacles are and X is a smoothed version to ensure that the encoded geometry is continuous satisfying Theorem 2. Then, the DAFNO
architecture uses the following instantiation of the kernel in (25) as
(x) f. x(=)x()) Ki ()(()) - v(x))dy.
(30)
where K,(æ y) is defined as the trigonometric polynomials as in the original FNO architecture. Lastly, we mention the subtraction of v(x)) v(y) (30), was introduced in You et (2022a) and has shown superior performance over FNOs.
C REVIEW OF VISCOSITY SOLUTIONS FOR PDES
Definition 2 (Viscosity solution (Katzourakis, 2014)). A bounded, uniformly continuous function V: X R is called a viscosity solution of the Fikonal initial-value Problem (7) provided
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i. V(z) = c(z) when E G
i. Given any function v E C2 (2), the following two hold
a. V u has a local maximum at the point x E X, then,
[|u((a))] c(x:) S 0.
(31)
V has a local minimum at the point X, then,
[[v(())] c(z) 2
(32)
We briefly mention that the existence of viscosity solutions can be shown in two ways namely Perron's 's method via the maximum principle or via the vanishing viscosity method. We briefly detail the latter result and refer the reader to Evans (2010) for more formal analysis.
Lemma 3. (Vanishing viscosity method Evans (2010)) Following the definition above, let Y be the viscosity solution to the given problem (7). Then, let Ve be the classical solution to the following viscous regularized Eikonal problem
[V(()c)(<z)I + Ee ((). =c(x),
Vc Fo, E Vg Eg.
(33a)
v(g) =c()),
(33b)
Then, Ve uniformly converges to as c +
D LEARNING VALUE FUNCTIONS: EXPERIMENTAL DETAILS AND ADDITIONAL RESULTS
D. A BRIEF REVIEW OF THE BASELINE METHODS
To evaluate our models, we aimed to compare across a series of different modern motion planning methodologies. In particularly, we consider perspectives across reinforcement like methods such as VIN and IEF, operator learning architectures such as FNO and DAFNO, and physics informed approaches in NTFields and P-NTFields. In this section, and testing. To clarify, all the 2D experiments use a NVIDIA 3090 Ti. The 3D iGibson experiments use NVIDIA A100 for data-generation and training, while employ a NVIDIA 4060 during testing.
series of different GPUs for training
Value Iteration Networks (VIN) Tamar et al. (2016) VIN attempts to learn value functions by approximating the value iteration (VI) algorithm. To do so, they introduce two neural network components to their algorithm. The first model, deemed the VI module, takes in the previous value function estimate and the current reward observed to provide an estimate the current value function. In implementation, for 2D experiments, the VI module is classical CNN. They then combine this module a planning module that takes the value function and current state and passes this through an attention mechanism coupled with classical NN to achieve the best path direction.
Implicit Environment Functions (IEF) Li et al. (2022) IN IEF2D/IEF3D the authors take the classical neural implicit network architecture, which represents 2D and 3D scenes via their signed distance functions, and instead learns to represent the scene by obtaining distances between samples start goal pairs. As such their methodology is continuous, and is able to learn trajectories by analyzing these distance functions. However, to generalize across environment's, the authors first project the scene to latent space via auto-encoder and then learn a neural implicit function in that latent space.
Fourier Neural Operators (FNO) Li et al. (2021) Like PNO, FNOs learn the solution of operator mappings across continuous function spaces. In particular, FNOs project the input map into a high dimensional latent space, of which an FFT is then performed. From here, the FNO learns weight matrix in frequency space that multiplies with the encoded input before performing an IFFT and then reprojecting back to the desired output resolution. They have been extremely successful in learning operator solutions specifically in the context of weather (See Kurth et al. (2023)).
Domain-Agnostie Fourier Neural Operators (DAFNO) Liu et al. (2023) DAFNO, an extension of FNO, maintains the same network structure as an FNO, but additionally encodes the domain geometry into the kernel calculation in frequency space. That is,
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DAFNO, multiplies the encoded input in frequency space by additional mask containing the domain geometry. This approach, taking advantage of the fact that Fourier transforms only operate periodic grids, enable users to learn FNO approximations non non-uniform geometries.
Neural Time Fields (NTFields) Ni & Qureshi (2023a) NTFields is similar to the operator learning architectures in that the network aims to learn an operator, but specifically they constrain that operator to be the solution operator of Eikonal PDE. particular, they
re-parameterize the value function as a factorized Eikonal equation V g) --= where T(
7 is learned via neural network. They then perform training with physics informed loss based on the satisfaction the Eikonal PDE enabling NTFields to learn paths without training data. As such NTFields is able to generalize to high dimensional environments, but is constrained as it needs retraining for each new environment geometry.
P-Neural Time Fields (P-NTFields) Ni & Qureshi (2023b) P-NTFields is extension to NTFields where the authors propose a progressive learning approach. Particularly, they modify the model training of NTFields by linearly increasing the speed field as the model trained for more epochs. This helps eliminate the local minima in the value function and thus better captures the geometry of the environment. However, like NTFields, P-NTFields is physics informed approach and therefore is limited as it also requires retraining for each new environment geometry.
D.2 SMALL-SCALE 2D MAZE EXPERIMENTS
To compare with state of the art value function predictors such VIN and IEF2D, we explore the efficacy a PNO on the small-scale 2D Grid-world experiments in Tamar al. (2016). We used the same dataset used to test the VIN framework of which IEF2D also employs.
Table Model parameters and performance met- rics for PNO over the GridWorld dataset. The PNO for 8 was smaller than that trained for 16 16 and 28 x 28.
PNO model number parameters
Avg.
Avg. La
Map size
relative training data
relative test data
The parameters and the relative errors of our model are presented in Table 4. To develop our model, it took approximately 40 minutes of training on a NVIDIA RTX 3090 Ti GPU.
8x8
238816
0.019
0.022
16 16
690432
0.031
0.035
28
x 28
690432
0.027
0.045
25
25
Path
Path
Start
Start
Goal
Goal
20
20
10
10
Figure 6: Two examples of PNO planning on 28 28 Grid-world dataset. Planning is done on the value function generated by PNO using classical gradient descent.
D.3 MOVINGAI I CITY EXPERIMENTS
In this experiment, we highlight the super-resolution capabilities the planning operator architecture. To do so, created custom 64 x 64 resolution dataset (will be publicly available) and trained the
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neural operator on the custom dataset. Note, this dataset consists of just objects rectangular grid as shown in the example in Fig. 2. Furthermore, our dataset did not include any of the city maps, but the synthetic object maps closely resemble similar structure the city maps. Our architecture consisted of 157808 parameters and we achieved an L2 relative error of 0. for both training and testing taking approximately 10 minutes for training on NVIDIA 3090 Ti GPU.
To showcase our the benefit of both the obstacle encoding and inductive bias, we perform comparisons across wide set of operator learning benchmarks. We briefly summarize both the training and testing error in Table 1. Note all the testing sets in each category are the same maps structurally, just scaled the target resolution via averaging. For the in in-distribution obstacle dataset, we consider 1k: maps with 10 different goals each for training. and 10 maps with 10 different goals each for testing. For the MovingAI city experiments, we include all 30 city maps with 3 goals each for testing. An example of the MovingAl city result along with comparison errors is given in Fig. 8 for snapshot of Paris. Additionally, we show the effect of the PINN loss in Fig. which certainly improves the gradient's 's residuals. Furthermore, to all models were trained and tested using the PyKonal FMM for the value function and a classic numerical solver for the SDF was used for DAFNO. For PNO, the FNO generated SDF was trained over 1k instances of the 64 x 64 dataset (achieving an relative L2 error of 0.068) and is also performing super-resolution when applied.
Lastly, for the same calculations, compute the speedup of the neural operator at various resolutions on a NVIDIA 3090 Ti GPU (MI. models) and utilize the extremely powerful AMD Ryzen 9 7950X CPU for the numerical solvers. In Table 5, we break down exactly the calculation time of both the SDFs as well as the value function where on small scaled maps, it is clear the numerical solver performs best; however, as the scale increases, we can see that the operator architectures do not lose much computation expenditure while the numerical solvers scale poorly.
Table 5: Computation times for super-resolution calculations average over 1000 instances on the Moving AI lab 2D dataset (642 indicates 64 x 64). The DAFNO SDF was calculated using the SciPy numerical solver. The numerical solver used for FMM is via Pykonal.
Avg. computation time
Avg. computation time
Arg. computation time
signed distance function( 1000 inst, ms)
value function 1000 inst, ms)
total function/ 1000 inst, ms)
1
Map size
64"
256
512"
10244
250
512"
1024"
250
512"
1024"
INO
16168
14336
L6563
2,281
16168
L4356
L4563
2.281
DAFNO
0.1734
19806
8,4007
46.1614
2,9232
2,9899
3,024
3,6626
3,0966
4.6695
11,4447
49.825
PNO
16168
1.3366
1.663
2.281
3,954
3.6729
4.6335
6.0332
5,5708
5.3085
6,3298
8.3142
Numerical
0.3454
5.9612
25.6555
104.6184
0.3454
5.9612
25.6555
104.6184
solver (FMM)
D.4 3D IGIBSON DATASET
For our 3D experiments, we use the iGibson Dataset Shen et al. (2021). The dataset consists of only 10 maps which are not sufficient train an accurate model. As such, we perform two augmentations for generating sufficient training data. First, we consider different training instances by rotating the different maps across about the z-axis by 90 degrees creating four versions of the same map. Additionally, we augmented the dataset with 32 maps from the HouseExpo dataset Li et al. (2020). The maps were extruded form 3D maps. Then, for each map, we randomly sampled 5 start goal pairs leading to a total dataset size of 360. We then performed a 90/10 train test split explicilly ensuring that the training dataset did not contain any Samuel environments. Further, no start goal positions in the evaluation dataset in Fig. 3 were seen during training. Fig 3 shows the best performing La example of our result. For completeness, we also present the worst performing example of our approach in Figure 9. Our model consisted of 418528 parameters of which Le relative training error and 0.19 L2 relative testing error for learning the value function. The computational calculations for value function generation were computed using a NVIDIA 4060 GPU.
achieved a 0.08
D.5 PLANNING WITH A 4 DOF MANIPULATOR
For our 4-DOF Manipulator experiments, we generate our dataset by randomly positioning the obstacles in the workspace of the robot. We choose the OpenManipulator-XX (RM-X52-TNNM, to test our methods. For training our PNO, generate the binary occupancy map in the configuration space of the robot of the size 1 We use 40 maps with 10 randomly generated goal positions each to generate our training data. The test set used 10 maps with 10 randomly generated goals each. To
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generate the binary occupaney map, we use the checkCollision function available in MATLAB for each state in the state space of the manipulator. Our model consisted of 55132 parameters. In terms achieved 0.027 L2 relative training enror and 0.041 L2 relative testing enror. The model was trained and tested on a NVIDIA 3090Ti GPU. The plan generated by our method was simulated and rendered MATLAB. 10 shows slice of of the value functions generated in the configuration space with Joint 3 and Joint 4 fixed at 0%. The Binary occupancy map generation took 5 hrs for 50 maps with randomly positioned obstacles. For training, FMM was used and the data generation took 40s while the model took 20 min to be trained over 200 epochs.
pf performance,
E EMPLOYING OPERATOR LEARNED VALUE FUNCTIONS AS NEURAL HEURISTICS: EXPERIMENT DETAILS AND ADDITIONAL RESULTS.
For this experiment, we trained directly on the city maps at 2562, 512² and 1024 resolutions. All of the training and testing in this section was completed using an NVIDIA A100. To build our dataset, we considered the 30 city maps provided by the MovingAl city dataset along with 10 goals for each randomly erode each of the maps between 1 and 30 layers generate more
map. Additionally, data and to help the operator infer the effect of erosion. Again, for each eroded map, we use 10 goals.
in- sufficient for generating training data. This is due the fact A * algorithm works using set of discrete control inputs to find a path with the shortest distance. However, the value function generated by the Eikonal equation (FMM) repre- sents a value function for a continuous control in- put space. Using this directly heuristic does not yield an accurate value function since such function largely under approximates the cost- to-go function for the shortest distance problem with discrete control space. In order to alleviate this issue train neural operator value functions generated using the Dijkstra Algorithm that gives the value function at every node for the same set of control inputs. This required a data generation time of hours for the 1024 x 1024 city maps (3 hours and 25 minutes for the 512 x 512 and 256 x 256 maps), but only needed to be complete once, offline. We provde the full model sizes as well the relative errors Table
For employment
heuristic, FMM
Table Model parameters and performance met- rics for the PNO models trained and deployed as neural heuristics.
PNO model
Avg. La
Avg. Le
Map size
number parameters
relative error training data 0.051
relative error
data
256 256
26029
0.065
512 512
102048
0.049
0.055
1024
161920
0.058
In addition to our results in Section 5, we also conducted an analysis on the effect of erosion. Fig. 11 shows that as the amount of layers eroded increases, the path improves at the cost of expanding more nodes as the heuristic which is as expected given that fully eroded map yields the Euclidean norm. Perhaps future work can explore different eroding methods for improving the admissibility of the value function.
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Value functions and relative error between FMM (Unseen)
PNO woout PINN
PNO PINN
FMM
Error PNO w/out PINN
PNO w/ PINN error
Gradient value function (I/VV and error
of gradient value function (x) c(x)) (Seen)
PNO w/out PINN
PNO w/ PINN
FMM
1.5
1.0
0.5
0.0
PNO w/out PINN Gradient Error
PNO w PINN Gradient Erro
1.00
0.75
0.50
Figure 7: Comparison of PNO with and without the hybrid PINN loss. The left image shows the value function and corresponding error while the right image shows the gradient norm [V and the corresponding gradient norm error. The example given is in the test set from the 64 x 64 synthetic maps and the red dot indicates the goal position. For this example, the model was trained with ę = 0.05 in (15).
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Value functions and relative errors (Paris, 1024x
FNO
DAFNO
1000
1000
750
500
500
250
500
750
1000
250
750
1000
PNO
FMM
1000
1000
600
500
20
400
200
500
1000
250
750
FNO Error
DAFNO Error
PNO Error
1000
1000
1000
750
750
750
500
500
500
2
250
250
0
250
500
750
1000
250
500
750
1000
250
500
750
1000
Figure 8: Example of various operator architectures Paris 1024 1024 map. The red dot indicates the target goal position.
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Bolton environment NTFields P-NTFields
FMM
PNO (ours)
Samuel environment NTFields
FMM
P-NTFields PNO (ours)
Figure 9: A second example of comparison between NTFields, P-NTFields, FMM and PNO unseen during training. This example is the worst-performing example for the PNO in terms of L2 loss.
Ground Truth
Value Function Approximation
15
15
Ground Truth
Value Function Approximation
Error
15
10
10
Figure 10: Value function approximation by PNO in 4D C-Space for examples shown in Fig. 4. The value functions are visualized as slices, with Joint 3 and Joint 4 fixed at an angle of 0%
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256 X 256
512 X 512
1024 X 1024
170.4
275
498
Avg Path Cost
170.1
274
496
169.8
273
494
169.5
272
492
5
0
10
15
10
15
Planning time (s) Nodes expanded
3300
9350
32000
3000
8800
30000
2700
8250
28000-
2400
7700
26000
5
10
15
0
5
10
15
10
15
0.20
0.59
2.06
0.18
0.55
1.94
0.16
0.51
1.82
0.14
0.47
1.70
15
10
Erosion
Erosion
Erosion
Figure 11: Effect of erosion on various parameters
25


Reviews:
[{"review_id": "L1Xn7pyTmP", "reviewer": "Reviewer_C6NW", "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.\n\nI 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.", "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.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "See review above", "weaknesses": "See review above", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9Rz2XGEMuD", "reviewer": "Reviewer_6Xqc", "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.", "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:\n\n1. 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.\n\n2. 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.\n\n3. 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.\n\n[1] Fourier Neural Operator with Learned Deformations for PDEs on General Geometries", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "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.\n\n2. 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.\n\n3. Computational Efficiency: The approach demonstrates fast computational performance for small-resolution cases, making it potentially suitable for real-time applications.", "weaknesses": "1. Scalability Constraints:\nSince 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.\n\n2. Dependence on Supervised Learning: \n The method’s reliance on supervised learning demands ground truth PDE solutions, which may be impractical to obtain in complex environments.\n\n3. Presentation Issues:\n - 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.\n- 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 $.\n\n4. Experimental Comparisons:\n 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.\n\n[1] Differentiable Spatial Planning using Transformers\n\n[2] Progressive Learning for Physics-Informed Neural Motion Planning\n\nThe 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hleVYRogy7", "reviewer": "Reviewer_hZWb", "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.", "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?\n- 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?\n- 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?\n- 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.\n- In table 1, do you have an explanation on why the success rate of your method particularly drops for the 16x16 maps ?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "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.\n- The authors provide a detailed literature review and clearly position the proposed approach within existing work  in traditional and neural motion planning.\n- 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.\n- 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.\n- 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)