Cross-Variable Loss Improves Multivariate Time Series Forecasting

Multivariate time series forecasting faces unique challenges because future variables often co-evolve under complex, shared system dynamics. While existing models typically focus on historical cross-variable dependencies, the relationships among future values are often underexplored. Modern forecasting models, largely following the Direct Forecasting (DF) paradigm, use point-wise objectives that do not explicitly enforce cross-variable structural constraints, leading to an "objective gap." To address this, researchers introduce Cross-Variable Loss (CvLoss), a novel plug-in structural regularizer. CvLoss works by constraining forecast residuals on a cross-variable graph, penalizing inconsistent edge-wise residual differences over forecast patches. This mechanism encourages consistency across both synchronous and asynchronous interactions between variables in the forecasted future. Experiments demonstrated that CvLoss consistently improves the performance of various competitive forecasting models. It outperformed representative learning objectives and proved compatible with a wide range of forecasting backbones. This indicates that explicitly modeling and constraining cross-variable dependencies in future predictions is crucial for enhancing the accuracy and reliability of multivariate time series forecasts.