This paper presents a method that uses neural networks as a caching mechanism to reduce the variance of Monte Carlo Partial Differential Equation solvers, such as the Walk-on-Spheres algorithm [Sawhney and Crane 2020]. While these Monte Carlo PDE solvers have the merits of being unbiased and discretization-free, their high variance often hinders real-time applications. On the other hand, neural networks can approximate the PDE solution, and evaluating these networks at inference time can be very fast. However, neural-network-based solutions may suffer from convergence difficulties and high bias. Our hybrid system aims to combine these two potentially complementary solutions by training a neural field to approximate the PDE solution using supervision from a WoS solver. This neural field is then used as a cache in the WoS solver to reduce variance during inference. We demonstrate that our neural field training procedure is better than the commonly used self-supervised objectives in the literature. We also show that our hybrid solver exhibits lower variance than WoS with the same computational budget: it is significantly better for small compute budgets and provides smaller improvements for larger budgets, reaching the same performance as WoS in the limit.