# Forecasting Multiple Observables with SCROLL: Score-Trained Uncertainty for Stochastic Dynamics

arXiv: https://arxiv.org/abs/2608.25898 · HTML: https://mlsanity.com/paper/2608.25898/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 13 references that are in the map.
- First posted on arXiv 26 Aug 2026.

## Builds on (most cited first, 6 of 6)

- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Multi-Task Learning Using Uncertainty to Weigh Losses for Scene Geometry and Semantics](https://mlsanity.com/paper/1705.07115/) (2017)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Variational Bayesian Last Layers](https://mlsanity.com/paper/2404.11599/) (2024)
- [Heterogeneous Multi-output Gaussian Process Prediction](https://mlsanity.com/paper/1805.07633/) (2018)

## Abstract (arXiv)

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.