Task-Locked Predictive Adaptation: Leftover Residual Densities from Few Target Labels

A trained neural operator is often reused when the unresolved forcing of a PDE changes. Users already rely on a predictive distribution for one operational functional, such as a regional mean. Recalibrating the residual as an unrestricted joint density moves that functional. Few labeled target fields are available. The leftover residual still requires a density. We keep one shared density for the task coordinate. The labels update only the density of the remaining coefficients given the task. From the source we build a dictionary of these leftover models: scale and tail changes, coupling to the task, and both together. Sequential likelihood updates the weights. The deployed leftover density averages the sequential predictors. We call this task-locked predictive adaptation. The adapted predictor has the same task law as the shared task density. Under i.i.d. target sampling, with the dictionary fixed from the source, the expected excess log loss relative to the source leftover density is at most $-\log ({\pi }_{\mathrm{ref}})/(K+1)$. Our analysis separates what the labels can buy from what the dictionary itself can represent. We test viscous Burgers with a frozen Fourier neural operator, ten training seeds, and 32 target labels. Residuals are scored on eight Fourier coefficients. Under a joint change of forcing tail and dependence, the leftover log score improves by 0.450 nats. Under no shift the mean change is 0.0005 nats. A maximum-likelihood fit of the same dictionary satisfies its calibration criterion in every recorded fit but scores 0.0717 nats higher on held-out fields. Composed corrections improve that score by 0.050 nats over a mixture of the separate families.