Planning under uncertain control execution noise demands reliable feedback policies that work across diverse environments. A robot that plans a path assuming perfect actuation will fail when its motors drift or its wheels slip. The problem grows harder in higher dimensions where configuration spaces balloon and traditional grid-based solvers become impractical. This paper introduces Planning Under Noise (PUN), a neural operator that learns noise-aware value fields for planning in unseen geometries. It builds on a screened-Poisson formulation of stochastic optimal control and tackles two core difficulties: specifying boundary conditions that support navigation across varying geometries and computing fields with trustworthy gradients at reasonable cost.
From screened-Poisson to neural operators: the theoretical foundation
The screened-Poisson equation arises naturally in stochastic optimal control when you model the expected cost-to-go under Gaussian execution noise. The value function satisfies a linear PDE whose solution gives the optimal feedback policy through its gradient. Classical approaches discretize the configuration space and solve the PDE numerically. That works in two or three dimensions but collapses in four or more because the number of grid points grows exponentially. Mesh-free Monte Carlo solvers avoid this curse by sampling paths instead of discretizing space. Walk on Stars (WoSt) is one such solver. It simulates stochastic trajectories that bounce off obstacles and terminate at the goal, then averages their costs to estimate the value at any query point. The method is exact in the limit of infinite samples and requires no mesh. However, each evaluation demands many samples to reduce variance, making it too slow for real-time replanning. PUN replaces the per-query Monte Carlo estimation with a learned mapping from geometry to value field, trained on noisy but informative Monte Carlo labels.
Why zero-Dirichlet boundaries fail and how Neumann conditions fix it
Standard formulations impose zero-Dirichlet conditions on obstacle boundaries: any path that touches an obstacle receives zero value and terminates. This seems intuitive — hitting a wall should be bad — but it creates a subtle problem. In a long corridor, the goal signal must propagate backward through many steps before it reaches the start. Each step attenuates the signal because paths that wander toward walls die early and contribute nothing. The result is a weak, noisy gradient far from the goal that struggles to guide the planner. The authors instead impose reflecting Neumann conditions on obstacles. A path that hits a wall reflects and continues exploring free space until it eventually reaches the goal. The Dirichlet condition remains only at the goal itself, where the value is fixed to zero. This change keeps the goal signal alive along arbitrarily long routes, strengthens gradient estimates in narrow passages, and makes the Monte Carlo estimator more reliable without requiring more samples.
Extending Walk on Stars to higher dimensions without a mesh
WoSt was originally designed for two-dimensional domains. The extension to four and higher dimensions requires careful handling of the stochastic process. The key insight is that the WoSt algorithm — which samples the next point uniformly on the boundary of the largest inscribed sphere centered at the current location — generalizes directly to any dimension. The sphere becomes a hypersphere, the boundary sampling uses standard high-dimensional uniform sampling techniques, and the walk proceeds identically. No configuration-space mesh is ever constructed. The solver evaluates the value at arbitrary points by launching independent walks and averaging their accumulated costs. The computational cost per sample grows modestly with dimension, but the number of samples needed for a given accuracy grows sharply because of variance. This variance is the bottleneck that motivates amortization.
Amortizing Monte Carlo computation with a neural operator
Rather than running WoSt to convergence for every test geometry, PUN learns to predict the value field directly from the geometry. The training data consists of pairs: a geometric description (obstacle locations, goal position) and a noisy, partially converged Monte Carlo estimate of the value field. The neural operator architecture processes the geometry as a continuous function — typically via a Fourier neural operator or similar architecture that handles resolution-invariant inputs — and outputs a scalar field defined over the domain. Training minimizes the difference between the predicted field and the Monte Carlo labels. Crucially, the labels need not be fully converged. Even noisy, early-stopped Monte Carlo estimates contain enough signal for the operator to learn the underlying geometry-to-field mapping. The operator effectively denoises the Monte Carlo output while learning the functional relationship. At training time, the authors generate diverse random geometries, run WoSt for a fixed budget of samples per geometry, and use those fields as supervision. The trained operator then produces a complete value field for a new geometry in milliseconds.
Test-time scaling through bounded local search
The predicted global field gives a strong initial policy, but local Monte Carlo noise in the training labels can cause small errors in the predicted gradient. To correct these, the authors add a test-time refinement step. Starting from the action suggested by the predicted gradient, the planner runs a bounded local search: it samples a small neighborhood of actions, evaluates each using the predicted field (or a few additional WoSt walks for higher fidelity), and selects the best. This search is bounded in both the number of candidates and the radius of exploration, so its cost is constant and small. The global field provides the coarse direction; the local search corrects fine-grained errors. This combination yields a feedback policy that is both fast and robust, with the local search absorbing the residual error from the neural operator's predictions.
Results across noise levels and dimensions
The evaluation spans planning tasks in two, three, and four dimensions across four noise levels. Baselines include sampling-based planners (RRT, PRM), optimization-based methods (MPPI, iLQR), and prior learning-based approaches. On the four-dimensional replanning benchmark, PUN achieves 95 to 100 percent success across all noise levels while maintaining the lowest runtime of any method. At the highest noise level in four dimensions, PUN improves success rate by 10.7 to 60.6 percentage points relative to the next best method and runs 3.29 to 32.15 times faster. The speed advantage comes from amortization: the neural operator evaluates in constant time regardless of geometry complexity, while sampling-based methods scale poorly with dimension and noise. The success rate advantage comes from the boundary condition change and the denoising effect of the neural operator, which together yield more reliable gradients in high-noise regimes where Monte Carlo variance would otherwise dominate.
Limitations: where the approach struggles
The method inherits constraints from its components. The reflecting Neumann boundary assumes obstacles are hard and perfectly reflective, which may not hold for soft constraints or uncertain obstacle geometry. The neural operator requires a training distribution that covers the test geometries; performance degrades on out-of-distribution shapes or topologies. The Monte Carlo labels used for training are biased at low sample counts, and while the operator learns to denoise them, systematic bias from early stopping can persist. The bounded local search can only correct errors within its search radius; if the predicted field points toward a local minimum far from the true gradient direction, the search may not escape. Finally, the approach assumes Gaussian execution noise; non-Gaussian or state-dependent noise would require a different PDE formulation and a retrained operator.
What this means for robotics and control engineers
For engineers building robots that must replan quickly under uncertainty, PUN offers a practical path forward. The neural operator can be trained offline on a cluster using simulated geometries that cover the expected deployment scenarios. At runtime, the model loads onto an onboard GPU and produces value fields in milliseconds. The bounded local search runs on CPU and adds microseconds. This enables closed-loop feedback at control frequencies — 100 Hz or higher — even in four-dimensional configuration spaces. The approach is especially valuable for systems with significant actuation noise (quadrotors in wind, manipulators with cable stretch, mobile robots on slippery terrain) where traditional planners either fail or require prohibitive computation. The code and training recipes are available from the authors, and the architecture integrates with standard robotics stacks (ROS 2, Drake, OMPL) through a simple service interface that accepts a geometry description and returns a value field or a policy function.