Recent work on the sequence universality of State Space Models (SSMs) has introduced efficient, maximally expressive continuous-time approaches for time-series modelling. While these works focus on discriminative settings, we extend this perspective to generative time-series modelling by proving that maximally expressive Structured Linear Controlled Differential Equations (SLiCEs) are universal time-series generators, in the sense that they can approximate the induced path laws of continuous causal pushforwards on compact latent sets in $W_\infty$. Building on these theoretical results, we propose Generative SLiCEs (G-SLiCEs), a maximally expressive continuous-time model for flow matching on path-space. Empirically, we show that expressivity improves performance in probabilistic forecasting and downstream tasks, while retaining the advantages of continuous-time models such as generalising to arbitrary observation grids. This is particularly beneficial for irregular grids, where fixed-grid models often struggle.

翻译：近期关于状态空间模型（SSMs）序列通用性的研究，为时间序列建模引入了高效且最大化表达的连续时间方法。尽管这些工作聚焦于判别式场景，我们通过证明最大化表达的结构化线性受控微分方程（SLiCEs）是通用时间序列生成器（即它们能在$W_\infty$中逼近紧致潜集上连续因果前推的诱导路径律），将这一视角扩展至生成式时间序列建模。基于这些理论结果，我们提出生成式SLiCEs（G-SLiCEs），一种用于路径空间流匹配的最大化表达连续时间模型。实验表明，表达性提升了概率预测及下游任务的性能，同时保留了连续时间模型的优势，例如可泛化至任意观测网格。这对于固定网格模型常遇困难的非规则网格尤为有利。