Shrinkage-Weighted Bellman Regression for Neural PDE Solvers

Random-walk PDE solvers fit neural networks to sampled Bellman targets whose noise varies across the domain. Weighting by return spread can control noisy updates, but the spread estimates are themselves uncertain and downweighting can weaken learning near steep solution features. We propose bounded shrinkage weighting for Bellman regression. The method accounts for sampling uncertainty in the estimated spread and shrinks weights toward uniformity when spatial differences are poorly resolved. Clipping and a uniform weight floor limit concentration and preserve a minimum contribution from every point. Our analysis distinguishes minimax noise-amplitude weighting from minimum-variance weighting, bounds the bias caused by dependence between weights and targets, and characterizes how reweighting changes a local least-squares fit. In a baseline study covering seven Burgers settings with 40 paired trials per setting, plain inverse-spread weighting lowers mean sampled hard-region error by approximately 42–71% relative to uniform regression, while individual runs can be less accurate. These results identify the practical challenge addressed by the safeguards: controlling noisy updates without losing useful corrections near sharp layers. The bounded-shrinkage comparisons and component ablations remain pending.