One Patch, Three Roles: What Is Actually Coupled in Autoregressive Time-Series Forecasting?

Adaptive patching is a recent and compelling proposal for time-series Transformers: allocate finer patches where the sequence looks locally informative. This paper asks under what conditions a content-adaptive patching operator should outperform a tuned uniform one. Local heterogeneity alone is not enough: under pointwise forecasting losses, a complex-looking region is not automatically one where finer patching reduces the loss. We model patching as a budgeted bitrate allocation and derive an explicit threshold that a dynamic patching rule must satisfy to beat a well-tuned uniform baseline, then bound the achievable improvement both locally (a quadratic surrogate) and globally (a strong-convexity bound under the model's assumptions). Two structural results follow: without a coupling constraint, scalar local complexity cannot produce a non-uniform optimum under a common loss landscape; and once the backbone is trained to its representation-aware optimum, the alignment gain collapses around a well-tuned uniform patch size. To test these predictions, we run a controlled isolation study on three representative architectures, replacing each adaptive mechanism with a uniform patch-size sweep while keeping the backbone, data, and training protocol fixed. On standard long-horizon forecasting benchmarks, the validation-selected uniform baseline is competitive with the dynamic counterpart, with per-setting effects concentrated near zero and no consistent directional advantage once results are aggregated by dataset. The larger gains we do observe are method- and dataset-specific. Adaptive patching should therefore be evaluated against a tuned uniform baseline; its value depends on whether a cheap and reliable routing signal can identify where finer patches actually reduce forecasting loss.

Time series forecasting serves as an essential tool for many real-world applications, supporting tasks such as resource optimization and decision-making. Despite significant architectural advancements, most modern models still treat forecasting task as a fixed mapping from history to target horizons. This induces temporal decoupling across future time points and limits the model's ability to adapt to the evolving context as forecasting progresses. In this work, we present LeapTS, a novel framework that reformulates time series forecasting as a dynamic scheduling process over the prediction horizon. Specifically, LeapTS organizes the forecasting process into multi-level decisions using: (1) the hierarchical controller to dynamically select the optimal prediction scale and advancement length at each step, and (2) continuous-time state evolution driven by neural controlled differential equations. Within this process, the controlled update mechanism explicitly couples the irregular temporal dynamics with discrete scheduling feedback. Extensive evaluations on both real-world and synthetic datasets demonstrate that LeapTS improves overall forecasting performance by at least 7.4% while achieving a 2.6$\times$ to 5.3$\times$ inference speedup over representative Transformer-based models. Furthermore, by explicitly tracing the scheduling trajectories, we reveal how the model autonomously adapts its forecasting behavior to capture non-stationary dynamics.

Residual error propagation remains a fundamental problem in recurrent models, where small prediction inaccuracies compound over time and degrade long-horizon performance. Accurately modeling the correlation structure of such residuals is critical for reliable uncertainty quantification in probabilistic multivariate timeseries forecasting. While recent time-series deep models efficiently parametrize time-varying contemporaneous correlations, they often assume temporal independence of errors and neglect spatial correlation across the observed network. In this paper, we introduce Teger, a structured uncertainty module that overcomes the spa- tial and temporal limitations of error-correlated autoregressive forecasting. Teger proposes a spatial curvature-aware graph rewiring mechanism explicitly strengthening information-bottleneck edges identified by discrete Forman curvature. The component is integrated into a low-rank-plus-diagonal covariance head, preserving tractable inference via the Woodbury identity. Teger is backbone-agnostic, requiring only the latent state produced by any autoregressive encoder. We provide theoretical evidence of Teger, and experimentally evaluate it on LSTM, Transformer, and xLSTM backbones across four real-world spatio-temporal datasets, showing consistent improvement in Continuous Ranked Probability Score (CRPS). We further provide a formal theoretical analysis connecting curvature-aware rewiring to (i) oversquashing alleviation, (ii) improved spectral connectivity, (iii) reduced effective resistance, and (iv) improved covariance calibration bounds