统一建模多个可观测变量的不确定性，提升预测精度与校准性。
Forecasting Multiple Observables with SCROLL: Score-Trained Uncertainty for Stochastic Dynamics
- 共享主干网络，通过自由路由最后一层信念联合建模多变量概率分布。
- 在异方差系统上实现最低负对数似然，校准性优于调参模型。
- 适用于需要多目标不确定性预测的场景，如气象与复杂动力系统。
预测随机动力系统通常不只关注单一数值，而是需要多个可观测变量——未来状态、阈值事件、状态标签——各自的概率分布。传统多任务方法通过平衡各任务损失来实现，需手动或学习调整权重。本文提出将各可观测变量的似然函数组合在共享主干网络的最后一层自由路由信念中，使单元相关的损失缩放自动融入可学习的似然参数，在同一梯度步内完成优化。随机动力系统提供了静态基准无法获得的可计算真实预测方差。理论预期得到验证：在设定良好的同方差奥尔施泰因-乌伦贝克过程上，模型恢复了分析解核函数；在异方差系统（随机洛伦兹-63、真实空气质量数据）中，信念模块能根据输入动态调节方差，单次运行即达到状态与状态任务的最佳负对数似然，校准性仅低于其表现更优的对手模型，且仅需极小的调参开销。在真实序列上，状态预测性能在五个滚动起点均保持优势。
原文摘要 · Abstract (English)
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.
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