Causal in Time, Circular in Space for High-Dimensional Time Series Forecasting

A multivariate time series records many variables over time, and a forecaster must learn relationships along both axes. Modern sensor networks, power grids and web traffic can contain thousands of variables, making attention across variables slow because its cost grows quadratically. A fast global convolution is a cheaper alternative, but it naturally wraps the end of an axis back to its beginning. Along time this hands a position trained to forecast what follows it part of its own target, but it is natural for a variable axis with no intrinsic first or last variable. We introduce Time Fluxer (TiF), which replaces attention with a fast, input-dependent global convolution, the fluxer, and gives the two axes different closures, causal along time and circular across variables. The causal closure lets every position be trained to forecast what follows it, so one model serves any history within its window; the circular closure gives no variable an end. On seven Time-HD datasets, replacing variable-wise attention (iTransformer) with the fluxer reduces error in 26 of 28 settings, and replacing patch-wise attention (PatchTST) reduces error in 21 of 28. A single model trained causally at every position also outperforms models trained separately for each shorter history in 26 of 35 dataset-history comparisons, whereas circular mixing along time reads its own targets during training and is worse than causal mixing at every shorter history length. Across variables, the fluxer scales as O(C log C) rather than the O(C^2) of attention, and in single precision it is faster than fused attention at every measured length. TiF therefore provides a fast and accurate way to model high-dimensional time series while supporting forecasts from any history within its window.