Zero-Shot Long-Term Forecasting in Foundation Models for Non-Autonomous Dynamical Systems

Most real world time series (TS) originate from some underlying dynamical system (DS), often with known external drivers or modulating factors like solar cycles or specific events. TS foundation models (TSFMs) can incorporate these drivers as covariates, but they are optimized for short-range forecasts and do not preserve a DS' long-term dynamical properties. By contrast, DS reconstruction (DSR) FMs learn and infer the TS-generating dynamical process, thereby predicting their long-term behavior. Existing DSRFMs are essentially autonomous, however, and do not naturally account for covariate information. Here we show that they can be enhanced by covariates through a simple trick, treating them as additional, driving state variables as in a skew-product conceptualization of non-autonomous DS. With this control-theory-inspired approach, we demonstrate that across multiple covariate TS forecasting tasks, our method matches or even surpasses state-of-the-art TSFMs, without any further modification of the architecture or retraining of the model. We also show that it can recover the limiting behavior (attractors) and forced dynamical regime changes that TSFMs miss. On motion-capture data, only DSRFMs, but not TSFMs, maintain realistic behavior over long time horizons.