CPiRi: Channel Permutation-Invariant Relational Interaction for Multivariate Time Series Forecasting

Current methods for multivariate time series forecasting can be classified into channel-dependent and channel-independent models. Channel-dependent models learn cross-channel features but often overfit the channel ordering, which hampers adaptation when channels are added or reordered. Channel-independent models treat each channel in isolation to increase flexibility, yet this neglects inter-channel dependencies and limits performance. To address these limitations, we propose $\mathbf{\text{CPiRi}}$, a $\mathbf{\text{channel}}\mathbf{\text{permutation}}\mathbf{\text{invariant}}\mathbf{\text{(CPI)}}$ framework that infers cross-channel structure from data rather than memorizing a fixed ordering, enabling deployment in settings with structural and distributional co-drift without retraining. CPiRi couples $\mathbf{\text{spatio-temporal}}\mathbf{\text{decoupling}}\mathbf{\text{architecture}}$ with $\mathbf{\text{permutation-invariant}}\mathbf{\text{regularization}}\mathbf{\text{training}}\mathbf{\text{strategy}}$: a frozen pretrained temporal encoder extracts high-quality temporal features, a lightweight spatial module learns content-driven inter-channel relations, while a channel shuffling strategy enforces CPI during training. We further $\mathbf{\text{ground}}\mathbf{\text{CPiRi}}\mathbf{\text{in}}\mathbf{\text{theory}}$ by analyzing permutation equivariance in multivariate time series forecasting. Experiments on multiple benchmarks show state-of-the-art results. CPiRi remains stable when channel orders are shuffled and exhibits strong $\mathbf{\text{inductive}}\mathbf{\text{generalization}}$ to unseen channels even when trained on $\mathbf{\text{only}}\mathbf{\text{half}}$ of the channels, while maintaining $\mathbf{\text{practical}}\mathbf{\text{efficiency}}$ on large-scale datasets. The source code is released at https://github.com/JasonStraka/CPiRi.