In computational tests, the new forecasting method SCROLL-MT followed changing uncertainty in a simulated stochastic system more closely when its final layer used the Full covariance option. On stochastic Lorenz-63 data, predicted variance had a Pearson correlation of 0.85 with a Monte Carlo estimate of the reference variance, compared with 0.61 for the Diag option. The contrast makes the covariance choice look important for this benchmark, but it was not a universal win: Diag had lower test likelihood scores than Full on all three PM2.5 tasks.
A model built for several kinds of forecast
SCROLL-MT jointly optimizes a shared deterministic backbone with task-specific likelihoods and free-routed Gaussian last-layer beliefs. The reported covariance variants are None, Diag and Full, with Diag as the default. Covariance is the part of the model that describes whether uncertainty in different outputs moves together. Each forecast combines a state or regression target, a binary event and an ordinal target at a fixed lead. The comparison included fixed or grid-tuned MAP losses, Kendall learned uncertainty weights, input-dependent regression variance, Laplace belief selection, deep ensembles and separately trained single-task models.
The evaluation used a common single hidden layer of width 50, validation-selected backbones and 10 seeds. The MAP weighting comparison used a 49-run grid. Training used Adam with a learning rate of 0.03, at most 5,000 full-batch steps and early stopping based on summed validation negative log-likelihood.
Three different tests of forecast uncertainty
The benchmark used three time-series corpora. The Ornstein-Uhlenbeck set contained 6,500 windows, each starting from an independent stationary draw, so it was independent and identically distributed by construction. The stochastic Lorenz-63 corpus used 65 parallel chains, a 2,000-step burn-in and 100-step decorrelation gaps before yielding 6,500 windows. The splits for these synthetic systems were random and non-overlapping.
The real-data arm used hourly Beijing PM2.5 and meteorology from 2010 to 2014. Gaps of up to 6 hours were linearly interpolated, windows touching longer gaps were dropped, and the remaining 1,680 windows used a 24-hour stride. Unlike the synthetic systems, the PM2.5 series had no computable ground-truth variance, and its split was chronological.
The difficult case separated the options
The simpler Ornstein-Uhlenbeck check served as a test of whether the method could recover known behavior. Every approach came within 1.5% of the benchmark's entropy floor, the reference level for the likelihood score, at leads of 0.25 or longer, and within 4.5% at the 0.1 lead. The reported forecast spread stayed within 2% of the analytic reference variance at every lead, while the estimated observation-noise component stayed within 4%. In plain terms, the free-routed uncertainty did not add substantial spurious spread where the expected behavior was known.
The Lorenz test was more demanding because the expected spread changed with the state. The predicted variance varied by less than 1.3-fold across a 284-fold range in the Monte Carlo reference variance. Pearson correlations were 0.61 for Diag and 0.85 for Full; rank correlations were 0.56 and 0.66, respectively. Those reference values came from simulation, so the precise correlations and range estimates carry Monte Carlo uncertainty.
Negative log-likelihood, or NLL, is a score of probabilistic fit in which lower values are preferred in these comparisons. At the headline settings, Full had lower NLL than Diag on all three Lorenz tasks: 2.230 versus 2.333 for the state forecast, 0.169 versus 0.170 for the event forecast, and 0.843 versus 1.032 for the ordinal forecast.
The real-series result was narrower
On the Beijing PM2.5 series, the pattern reversed. Diag recorded NLLs of 1.250, 0.450 and 1.203 for the state, event and ordinal tasks, compared with 1.255, 0.489 and 1.212 for Full. A five-origin rolling analysis also found that Diag's state-head advantage over every weighting baseline ranged from 0.061 to 0.088 nats, with each difference at least 4.8 between-origin standard errors. The event head tied, while the regime head did not separate from the Kendall-CE baseline. A reported tie is not evidence of equivalence.
Accuracy was not the only difference
On Lorenz, the two SCROLL-MT variants also had much lower regression calibration errors than the tuned MAP grids. Diag and Full recorded errors of 0.016 and 0.015, respectively, versus 0.100 for the cross-entropy grid and 0.103 for the probit grid. Their wall-clock times were 18 and 29 seconds per seed, compared with 356 and 471 seconds for those grids; the grid-row times include tuning.
A self-tuned prior configuration came within 0.006 nats of the grid-selected regression NLL on every corpus and within 0.017 nats on every head, while using one prior configuration per backbone rather than seven. The result was not independent of prior strength: a strong prior cost 0.12 nats on the Lorenz ordinal head and doubled calibration error.
Joint training was not uniformly helpful. Relative to separately trained heads, it produced Lorenz state and regime interference of +0.057 ± 0.009 and +0.072 ± 0.024 nats. On PM2.5, the corresponding differences were -0.006 ± 0.007 for state, -0.018 ± 0.006 for event and -0.014 ± 0.004 for regime. The pattern shows that sharing a backbone can impose a cost on some tasks while helping others in the tested corpora.
A result bounded by its benchmarks
The work is an arXiv preprint, version 1, dated 26 August 2026. Its evidence covers two synthetic stochastic systems and one Beijing PM2.5 series. The real-data arm lacks a computable ground-truth variance, while the Lorenz variance targets are Monte Carlo estimates. The authors therefore present the findings as comparative evidence within these tests, not as a general guarantee that Full or Diag will perform best elsewhere.
Within the reported results, Diag is presented as the practical default for the real corpus, while Full is the stronger choice when the Lorenz benchmark benefits from a covariance family that can express more of its changing spread. Further tests would be needed before extending that trade-off beyond the systems and task inventory used here.
Paper data and sources
Original title: Forecasting Multiple Observables with SCROLL: Score-Trained Uncertainty for Stochastic Dynamics
Authors: Pavel Prochazka
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text