Universal Time Series Generation with Neural Controlled Differential Equations

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.

Approximate equivariance via projection-based regularisation

Equivariance is a powerful inductive bias in neural networks, improving generalisation and physical consistency. Recently, however, non-equivariant models have regained attention, due to their better runtime performance and imperfect symmetries that might arise in real-world applications. This has motivated the development of approximately equivariant models that strike a middle ground between respecting symmetries and fitting the data distribution. Existing approaches in this field usually apply sample-based regularisers which depend on data augmentation at training time, incurring a high sample complexity, in particular for continuous groups such as $SO(3)$. This work instead approaches approximate equivariance via a projection-based regulariser which leverages the orthogonal decomposition of linear layers into equivariant and non-equivariant components. In contrast to existing methods, this penalises non-equivariance at an operator level across the full group orbit, rather than point-wise. We present a mathematical framework for computing the non-equivariance penalty exactly and efficiently in both the spatial and spectral domain. In our experiments, our method consistently outperforms prior approximate equivariance approaches in both model performance and efficiency, achieving substantial runtime gains over sample-based regularisers.