MixRoute: Mixed Parameterization with Adaptive Information Routing for Time Series Forecasting with Exogenous Variables

Time series forecasting with exogenous variables (TSF-X) predicts an endogenous target series from its own history and from exogenous variables whose future values may be available in advance. The task therefore couples temporal dependence within the endogenous series with cross-variable dependence on the exogenous variables. Decoupled methods model these dependencies in separate modules and rely on their combination to capture correlations that span time and variables jointly. Joint methods instead model both dependencies in one token space, but parameter sharing and information routing remain important design considerations. Homogeneous parameterization applies the same transformations to endogenous patches and exogenous-variable tokens despite their different semantic roles. A single attention distribution per head couples temporal modeling with exogenous information acquisition, without explicit specialization for either role. To address these issues, we propose MixRoute, a unified Transformer framework that makes representation and information flow type-aware for endogenous patches, exogenous variables, and learnable global tokens. Under Mixed Parameterization, endogenous patches share one set of parameters, whereas each exogenous and global token has a private set. Adaptive Information Routing (AIR) uses separate queries for an intrinsic stream over the endogenous history and a cross-type stream over the exogenous and global tokens. It then combines the stream outputs with a gate computed from both streams. On 12 real-world datasets in the with-future setting, MixRoute reduces MSE by 8.0% on average relative to the strongest baseline evaluated. When future exogenous values are unavailable, MixRoute needs no architectural change and still ranks first overall.