Planning under noisy control execution usually requires feedback policies that work across many environments. The authors propose Planning Under Noise (PUN), a neural operator that learns noise-aware value fields for stochastic optimal control. The method builds on a screened-Poisson PDE formulation and avoids meshing the configuration space.
A key design choice is boundary conditions. Zero-Dirichlet obstacle boundaries terminate sampled paths at walls, which weakens the goal signal along long routes. PUN instead imposes reflecting Neumann conditions on obstacles and a Dirichlet condition at the goal, so sampled paths keep exploring until they reach the target. To solve the PDE without a mesh, the authors extend the Monte Carlo solver Walk on Stars to higher dimensions.
Because local Monte Carlo fluctuations can reverse gradient estimates and cause planning failures, direct planning would require costly sampling. PUN amortizes this by learning geometry-to-field mappings from noisy, partially converged Monte Carlo labels. At inference, test-time scaling uses bounded local search guided by the predicted global field.
In experiments across dimensions and four noise levels, PUN matches or outperforms all compared planners. On the 4D replanning benchmark it reaches 95-100% success, and at the highest tested noise level it shows 10.7-60.6% relative gains in success rate while running 3.29-32.15 times faster.