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 $16\times$ 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-DOF 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 ($A^\ast$, $RRT^\ast$).

翻译：本文提出了一种用于预测运动规划问题价值函数的规划神经算子（PNO）。我们将价值函数近似重新定义为从成本函数空间到价值函数空间学习一个单一算子，该算子由Eikonal偏微分方程（PDE）定义。因此，我们的PNO模型尽管是在粗分辨率下使用有限数量的样本进行训练的，却继承了神经算子零样本超分辨率的特性。我们在MovingAI实验室的2D城市数据集上展示了在训练分辨率$16\times$下的精确价值函数近似，与基于iGibson建筑数据集的3D场景上最先进的神经价值函数预测器进行了比较，并展示了4自由度机械臂的最优规划。最后，我们研究了将PNO输出的价值函数作为启发式函数来加速运动规划。我们从理论上证明，通过引入一个归纳偏置层来保证我们的价值函数满足三角不等式，PNO启发式函数是$\epsilon$一致的。使用我们的启发式函数，与经典规划方法（$A^\ast$、$RRT^\ast$）相比，我们在MovingAI实验室2D城市数据集上实现了访问节点数减少$30\%$，同时获得接近最优的路径长度。