Forecasting Multiple Observables: Score-Trained Uncertainty for Stochastic Dynamics

Forecasting a stochastic dynamical system rarely means a single number: one wants several observables-future state, threshold event, regime label-each with its own likelihood. Standard multi-task recipes balance per-task losses, tuned or learned. We instead compose the observables' likelihoods in per-task score-trained last-layer variance heads on a shared backbone; this absorbs unit-dependent loss scaling into per-task likelihood parameters learned in the same gradient pass. Stochastic dynamics supply what static benchmarks cannot: computable ground truth for the predictive variance. On the well-specified, homoscedastic Ornstein-Uhlenbeck process, as the proper-score argument predicts, the learned predictive law recovers the analytic kernel and correctly specified baselines tie. On heteroscedastic systems (stochastic Lorenz-63, real air-quality data) the input-dependent variance separates: best single-run NLL on the state and regime tasks (Full on Lorenz, Diag on the real series), and the gain is the score-trained variance's, regularised or not, not the weights'. On the real series the state margin holds across five rolling origins.