CoRe: Coherence and Relational Alignment for Multivariate Time Series Forecasting
Authors: Xiaoyu Lin, Huiran Duan, Yining Liu, Zhixiang Wu, Chu Lin, Lin Lu
Abstract
Direct forecasting has become a standard paradigm for multivariate time-series forecasting because it predicts the full future horizon in a single pass. However, its training objective is often still decomposed into pointwise errors such as MSE. Such objectives provide stable supervision, but they do not explicitly preserve the structure of the future trajectory: temporal coherence within each variable and relational consistency across variables can both be weakened. We propose CoRe, a model-agnostic learning objective for direct multivariate forecasting. CoRe replaces pointwise supervision with two output-space constraints: a frequency coherence loss that aligns predicted and target spectra, and a low-rank relational graph loss that matches sampled pairwise differences in a target-derived PCA subspace. The resulting objective introduces no trainable parameters and can be applied to existing forecasting backbones by changing only the loss. Experiments on standard benchmarks show that CoRe improves strong baselines, compares favorably with recent forecasting objectives, and remains effective across different backbones, datasets, and hyperparameter settings overall consistently.
Multivariate time series forecasting presents unique challenges because future variables often co-evolve under shared system dynamics. While existing studies mainly focus on cross-variable dependencies in historical observations, dependencies among future values are much less explored. Specifically, modern forecasting models largely follow the Direct Forecasting (DF) paradigm, generating multi-step forecasts with point-wise objectives that do not explicitly constrain cross-variable structure. In this work, we show that the DF objective is mismatched in the presence of cross-variable and lagged dependencies, revealing an objective gap. To address this issue, we propose \textbf{C}ross-\textbf{V}ariable \textbf{Loss} (CvLoss), a plug-in structural regularizer that constrains forecast residuals on a cross-variable graph. CvLoss penalizes inconsistent edge-wise residual differences over forecast patches, encouraging consistency across both synchronous and asynchronous interactions. Our experiments show that CvLoss consistently improves competitive forecasting models, outperforms representative learning objectives, and is compatible with a variety of forecasting backbones.
In data-driven training, multivariate time-series forecasting is usually optimized with a scalar loss averaged over samples, variables, and horizons. This averaging is convenient, but the optimizer sees only the aggregated gradient, which does not reveal whether the variable-wise contributions align or oppose one another. To quantify how often this disagreement arises, we measure the variable-wise gradients directly and find that 30.6% of their pairwise cosine similarities are negative on average across seven datasets. However, conflict and harm are not the same thing. Under shared training 35 of the 64 variables do worse than a full-input single-target oracle, and the harmed fraction is not reliably predicted by how often gradients conflict. We propose Per-Variable Surgery (PV-Surgery), an optimizer-side training strategy for backbones with cache-compatible layers. One backward pass builds variable-wise gradient proxies from output-side signals and keeps the pointwise forecasting loss. Reliability-aware selection targets layers whose proxy sums closely approximate their shared-gradient slices. Conditional pooling forms anchor and conflict pools without dropping variables. Common-direction surgery aligns variable or pooled gradients with their normalized mean and restores input norms to avoid reweighting. In experiments across five backbones, seven datasets, and four horizons, PV-Surgery lowers MSE by 3.61% and MAE by 2.93% on average. For multivariate forecasting, this indicates that the variable-wise structure hidden by mean-loss training is a usable optimization signal.
We introduce TiRex-2, a recurrent xLSTM-based time series foundation model that generalizes the univariate TiRex to multivariate forecasting with both past and future covariates. Real-world forecasting is inherently sequential: observations arrive continuously, variables evolve jointly, and a subset of covariates is known ahead of time. Existing Transformer-based time series foundation models capture cross-variate dependencies but incur quadratic complexity in context length and require full-history recomputation as new observations arrive. TiRex-2 addresses these limitations through a memory-centric recurrent design that operates at constant per-patch cost under streaming. The model combines a bidirectional time mixer with an asymmetric grouped-attention variate mixer, enabling the integration of future-known covariates while preserving strict causality over target variables. To our knowledge, this is the first time series foundation model that achieves this combination of properties. To support scalable multivariate pretraining, we propose a synthetic coupling pipeline that composes diverse multivariate samples on the fly from large univariate corpora. Empirically, TiRex-2 achieves state-of-the-art zero-shot performance on GIFT-Eval and fev-bench, remains stable when streamed to arbitrary context lengths, and maintains constant inference cost per patch. The model uses 38.4M active parameters in univariate mode, with an additional 44.1M parameters activated for multivariate forecasting.