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This paper introduces SCROLL, a method for forecasting multiple observables in stochastic dynamical systems by composing likelihoods through free-routed last-layer beliefs on a shared backbone. The approach effectively captures unit-dependent loss scaling within a single gradient pass, enabling accurate predictions across various tasks, including future state and regime labeling. Results demonstrate that SCROLL achieves superior performance in both well-specified and heteroscedastic systems, outperforming traditional methods while reducing computational costs.
SCROLL achieves best-in-class predictive accuracy for multiple observables in stochastic systems while cutting computational costs significantly.
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 free-routed last-layer beliefs on a shared backbone; this absorbs unit-dependent loss scaling into likelihood parameters learned in the same gradient pass. Stochastic dynamics supply what static benchmarks cannot: computable ground truth for the predictive variance. Results land where theory puts them: on the well-specified, homoscedastic Ornstein--Uhlenbeck process the learned predictive law recovers the analytic kernel and correctly specified baselines tie. On heteroscedastic systems (stochastic Lorenz-63, real air-quality data) the belief's input-dependent variance separates: best single-run NLL on the state and regime tasks, calibration matched only by arms whose NLL it beats, at a fraction of the tuned grids'cost. On the real series the state margin holds across five rolling origins.
