Reducing Noise in Monte Carlo Labels with Local Conservation

Machine-learning surrogates are often trained on Monte Carlo (MC) labels, whose sampling noise around the simulator's mean output is costly to reduce. A local conservation law, or balance, can help correct such a label after the run, but the label's mismatch with the balance contains both sampling noise and a persistent approximation error. We analyze a classical linear correction that uses the noise covariance and a trust level per equation to subtract the noise predicted from this mismatch. A first limit is that, when the balance is treated as exact, merging its local equations into coarser ones cannot increase the noise variance removed. A second is that, for a fixed simulator input, the label improves on average over runs exactly when this variance exceeds the squared bias from the persistent error. Since more samples shrink only the noise, one balance can improve a noisy label and degrade a cleaner one. Because this bias is unknown, a one-label diagnostic estimates a lower bound on the expected error reduction, but the estimate is noisy and does not guarantee improvement for a single label. In airport-dispersion tests at a nominal 300k particles per run, the correction improves 91% of the 34 evaluated labels against separate reference runs, on selected cells within 0–3 m of the ground. Estimating the variances and trust levels requires an extra, larger run, so the gain is in accuracy, not simulation cost. In separate experiments with more samples per run, the correction degrades many labels, as the second limit allows.