Planning under Noise with Neural Operators

Michael Chen; Hong Chul Nam; Xi Deng; Anima Anandkumar

cs.LGOct 20, 2026

Planning under Noise with Neural Operators

Authors: Michael Chen, Hong Chul Nam, Xi Deng, Anima Anandkumar

Organizations: Computing and Mathematical Sciences, California Institute of Technology · Center for Data Science, New York University

Abstract

Planning under uncertain control execution noise requires reliable feedback policies across diverse environments. We introduce Planning Under Noise (PUN), a neural operator that learns noise-aware value fields for planning in unseen geometries. Building on an established screened-Poisson formulation of stochastic optimal control, we address two challenges: specifying boundary conditions that support navigation across varying geometries and efficiently computing fields with reliable gradients. Zero-Dirichlet obstacle boundaries terminate sampled paths at walls, weakening the goal signal along long routes and making it harder to estimate reliably. We instead impose reflecting Neumann conditions on obstacles and a Dirichlet condition at the goal, allowing sampled paths to continue exploring free space until they reach the target. To solve the resulting PDE without a configuration-space mesh, we extend a Monte Carlo (MC) solver, Walk on Stars (WoSt), to higher dimensions. However, local MC fluctuations can reverse estimated gradient directions and cause planning failures, making direct planning require costly sampling. PUN amortizes this computation by learning geometry-to-field mappings from noisy, partially converged Monte Carlo labels, avoiding expensive convergence for every training example. At inference, test-time scaling uses bounded local search guided by the predicted global field. We evaluate PUN on planning tasks across dimensions. Across four noise levels, PUN matches or outperforms all compared planners, achieving 95-100% success on the 4D replanning benchmark while maintaining the lowest runtime. At the highest tested noise level in 4D, PUN achieves 10.7-60.6% relative gains in success rate over other methods while running 3.29-32.15 times faster.

Figures & tables

Fig. 1: (a) Fast replanning. PUN has the lowest mean runtime on unseen 4D maps: 10.8 ms/start, compared with 32.8–387.1 ms/start for the baselines. Runtimes are averaged equally over the four execution-noise levels in Table 5 and plotted on a logarithmic axis. (b) Resolution transfer. FNO maintains over 90% local-search success across all five resolutions without retraining, reaching 92.50% at $4096\times 4096$ after training at $256\times 256$ . Missing points indicate out-of-memory failures. (c, d) Continued goal seeking. In the illustrated geometry, the solution induced by the Dirichlet boundary cannot propagate goal information further since most trajectories are absorbed by obstacles. Consequently, it is more difficult for planners to find the target than with the mixed boundary condition.

Panel of Figure 2 — Fig. 2: Planning Under Noise: (A) Planning: a neural operator maps geometry, goal, cost, and noise inputs to a desirability field. Grid-based local search selects feasible routes with high endpoint desirability, executes a short prefix, and replans. (B) Training: Walk-on-Stars (WoSt) estimates provide pointwise supervision; neural caching supplies learned continuation at the walk horizon.

Figure 3

Fig. 3: Replanning trajectories on the same 4D manipulator, start, and goal at $\lambda=0.25$ (top) and $\lambda=2$ (bottom). Columns show successive execution states; PUN’s trajectory is orange. PUN reaches the goal in both illustrated rollouts. Aggregate success and runtime are reported in Table 5 .

Model	Gradient (%)	Local search (4) (%)	Local search (10) (%)	Relative $L_{2}$
UPT	2.48	2.78	3.57	0.5533
Transolver	3.26	5.34	15.30	0.5158
WoSt	2.71	6.30	36.45	0.1852
GINO	46.56	51.24	53.11	0.2628
U-Net	74.21	84.22	88.98	0.1605
DISCO	82.49	87.60	91.08	0.1041

Table 1: 2D application-benchmark evaluation of neural backbones trained with WoSt supervision and direct WoSt inference: planning success and relative

L_{2}

error against the FEM ground-truth field.

Table 6

Dim.	Method	$\lambda=0.25$	$\lambda=1$	$\lambda=2$	$\lambda=4$
2D	A*	59.00 / 43.63	42.33 / 47.09	35.00 / 51.18	23.67 / 53.76
	Dijkstra	59.00 / 154.94	42.33 / 174.95	35.00 / 203.41	23.67 / 239.97
	RRT	82.00 / 98.45	68.67 / 103.87	52.33 / 112.27	36.67 / 124.71
	MPPI	84.00 / 1734.11	61.00 / 1773.31	45.67 / 1838.73	31.67 / 2126.12
	PI 2 -CMA	78.33 / 1330.76	60.00 / 1484.91	48.00 / 1524.70	35.00 / 1606.82
	PUN (zero-shot)	91.67 / 9.55	74.00 / 11.14	62.33 / 14.07	45.67 / 14.51

Table 5: Replanning under execution noise on 2D and 3D application benchmarks and 4D manipulators. Each entry reports success (%) / runtime (ms/start). Bold marks the highest success and lowest runtime within each dimension and noise level, including ties.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Definition	Meaning
Control problem
$\xi,\ x$	§ 2.1	starting point of the process or walk; conditioned as $\mid\xi=x$ where the start is not otherwise fixed
$X^{\xi,\mathbf{u}}_{t}$	Eq. 1	controlled process at time $t$ , from the prescribed start $\xi$ under control $\mathbf{u}$
$\mathbf{b}$	Eq. 1	drift of the controlled system ( $\mathbf{b}=0$ in the specialization)
$G,\ R$	Eq. 1 , Eq. 2	control matrix and control penalty ( $G=R=I$ in the specialization)
$\Sigma$	Eq. 1	diffusion tensor; $\Sigma=\lambda I$ when $G=R=I$ under matching

Appendix

Table 6: Notation, with the equation, assumption, or section where each symbol is defined.

$\rho/r$	$0$	$1/6$	$1/3$	$1/2$	$2/3$	$5/6$	$1$
$\kappa r=1$	1.0000	0.9880	0.9593	0.9232	0.8880	0.8614	0.8509
$\kappa r=4$	1.0000	0.8558	0.6157	0.4086	0.2629	0.1766	0.1466

Appendix

Table 7: Star-step weight

Q^{\kappa}_{3}(\rho,r)

r=1

. The weight is

1

for a zero-length step and decreases with the realized hit distance

\rho

, reaching

\gamma_{3}(\kappa r)=\kappa r/\sinh(\kappa r)

\rho=r

. A step cut short by an obstacle loses less weight than a full sphere step.

$d$	$G_{\mathbb{R}^{d}}(r)$	$G^{D}_{B_{R}}(r)$	$\gamma_{d}(z)$
$2$	$\dfrac{K_{0}(\kappa r)}{2\pi}$	$\dfrac{1}{2\pi}\!\left[K_{0}(\kappa r)-\dfrac{K_{0}(\kappa R)}{I_{0}(\kappa R)}I_{0}(\kappa r)\right]$	$\dfrac{1}{I_{0}(z)}$
$3$	$\dfrac{e^{-\kappa r}}{4\pi r}$	$\dfrac{1}{4\pi r}\!\left[e^{-\kappa r}-e^{-\kappa R}\dfrac{\sinh(\kappa r)}{\sinh(\kappa R)}\right]$	$\dfrac{z}{\sinh z}$
general	$\dfrac{\kappa^{\nu}}{(2\pi)^{d/2}}\dfrac{K_{\nu}(\kappa r)}{r^{\nu}}$	$\dfrac{\kappa^{\nu}}{(2\pi)^{d/2}r^{\nu}}\!\left[K_{\nu}(\kappa r)-\dfrac{K_{\nu}(\kappa R)}{I_{\nu}(\kappa R)}I_{\nu}(\kappa r)\right]$	$\dfrac{(z/2)^{\nu}}{\Gamma(\nu+1)I_{\nu}(z)}$

Appendix

Table 8: Screened Poisson kernels by dimension. All entries follow from Eq. 17 , Eq. 18 , and Eqs. 20 – 21 with

\nu=d/2-1

Dim.	Method	$\lambda=0.25$	$\lambda=1$	$\lambda=2$	$\lambda=4$
2D	A*	[55.92, 62.08]	[39.16, 45.50]	[31.90, 38.10]	[20.89, 26.45]
	Dijkstra	[55.81, 62.19]	[39.15, 45.51]	[31.99, 38.01]	[21.08, 26.26]
	RRT	[79.37, 84.63]	[65.68, 71.66]	[49.14, 55.52]	[33.46, 39.88]
	MPPI	[81.70, 86.30]	[57.77, 64.23]	[42.45, 48.89]	[28.82, 34.52]
	PI 2 -CMA	[75.76, 80.90]	[56.81, 63.19]	[44.76, 51.24]	[31.95, 38.05]
	PUN (zero-shot)	[89.90, 93.44]	[71.27, 76.73]	[59.15, 65.51]	[42.55, 48.79]

Appendix

Table 9: 95% confidence intervals for replanning success rates (%) corresponding to Table 5 .

Dim.	Method	$\lambda=0.25$	$\lambda=1$	$\lambda=2$	$\lambda=4$
2D	A*	[24.99, 62.27]	[26.25, 67.93]	[28.10, 74.26]	[29.44, 78.08]
	Dijkstra	[79.37, 230.51]	[88.65, 261.25]	[102.34, 304.48]	[123.40, 356.54]
	RRT	[69.41, 127.49]	[72.10, 135.64]	[78.18, 146.36]	[82.46, 166.96]
	MPPI	[1227.92, 2240.30]	[1257.88, 2288.74]	[1279.80, 2397.66]	[1400.54, 2851.70]
	PI 2 -CMA	[964.06, 1697.46]	[1053.67, 1916.15]	[1067.48, 1981.92]	[1079.98, 2133.66]
	PUN (zero-shot)	[7.79, 11.31]	[8.80, 13.48]	[10.71, 17.43]	[10.19, 18.83]

Appendix

Table 10: 95% confidence intervals for replanning runtime (ms/start) corresponding to Table 5 .

Test set	Gradient	Local search (10)
3D application benchmarks	65.95	89.46
4D manipulators	97.50	99.48

Appendix

Table 11: FNO (WoSt) planning success (%) with gradient following and local search.

Method	2D zero-shot	4D manipulators
WoSt + FNO	1.20	2.79
Direct WoSt	1128.60	160.59

Appendix

Table 12: Full test-time cost (ms/start) on the 2D zero-shot application-inference subset and 4D manipulator test set. Both methods use local search in 2D and in 4D.

Fig. 4: Zero-shot FNO field predictions and extracted paths at $128\times 128$ , $200\times 200$ , and $256\times 256$ (left to right). Color shows predicted desirability $\Psi_{\theta}$ , gray denotes obstacles, circles mark starts, and the star marks the shared goal in each map.

Panel of Figure 16 — Fig. 5: Goal-conditioned field prediction and reuse across initial states.

Panel of Figure 17 — Fig. 6: WoSt walk and label statistics: (a) walk-length variability and boundary treatment; (b) label and gradient variability on a fixed map, with differences measured against a finite Monte Carlo reference; (c) spatial variation in mean walk length.

Fig. 7: Local search on one fixed urban-map field: (A) full map and routes; (B,C) enlarged neighborhoods for local search (4) and (10). Greater depth exposes endpoints farther along the passage. Gray cells are obstacles; overlays mark candidate regions and routes. Table 1 reports aggregate controller success.

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We introduce Neural Navigation Functions (Neural-NF), a learned reactive navigation function capable of zero-shot transfer across unseen environment geometries. Neural-NF places data-driven adaptation within a structured elliptic planner, where the navigation objective is learned while planner structure is preserved by construction. Specifically, intrinsic Laplacian-derived features are mapped to local PDE coefficients, and solving the resulting boundary value problem produces a globally consistent value function on each target domain. For every admissible learned model, the resulting policy is collision-free, provides monotonic descent and a global minimum at the goal by construction. This admits a linearly-solvable optimal-control interpretation for any parameter setting. Empirically, Neural-NF achieves strong zero-shot transfer across diverse geometries and outperforms learned planners that directly predict the value function by up to a

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