In multivariate time series forecasting, the most popular strategy for modeling the relationship between multiple time series is the construction of graph, where each time series is represented as a node and related nodes are connected by edges. However, the relationship between multiple time series is typically complicated, e.g. the sum of outflows from upstream nodes may be equal to the inflows of downstream nodes. Such relations widely exist in many real-world scenarios for multivariate time series forecasting, yet are far from well studied. In these cases, graph might be insufficient for modeling the complex dependency between nodes. To this end, we explore a new framework to model the inter-node relationship in a more precise way based our proposed inductive bias, , where a group of functions parameterized by neural networks are learned to characterize the dependency between multiple time series. Essentially, these learned functions then form a “field”, i.e. a particular set of constraints, to regularize the training loss of the backbone prediction network and enforce the inference process to satisfy these constraints. Since our framework introduces the relationship bias in a data-driven manner, it is flexible and model-agnostic such that it can be applied to any existing multivariate time series prediction networks for boosting performance. The experiment is conducted on one toy dataset to show our approach can well recover the true constraint relationship between nodes. And various real-world datasets are also considered with different backbone prediction networks. Results show that the prediction error can be reduced remarkably with the aid of the proposed framework.

中文翻译：

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函数关系域：多元时间序列预测的模型无关框架


在多元时间序列预测中，对多个时间序列之间的关系进行建模最流行的策略是构建图，其中每个时间序列表示为一个节点，相关节点通过边连接。然而，多个时间序列之间的关系通常很复杂，例如上游节点的流出总和可能等于下游节点的流入。这种关系广泛存在于多元时间序列预测的许多现实场景中，但还远未得到充分研究。在这些情况下，图可能不足以对节点之间的复杂依赖关系进行建模。为此，我们探索了一种新的框架，以基于我们提出的归纳偏差以更精确的方式建模节点间关系，其中学习一组由神经网络参数化的函数来表征多个时间序列之间的依赖性。本质上，这些学习到的函数形成一个“场”，即一组特定的约束，以规范主干预测网络的训练损失并强制执行推理过程以满足这些约束。由于我们的框架以数据驱动的方式引入了关系偏差，因此它非常灵活且与模型无关，因此可以应用于任何现有的多元时间序列预测网络以提高性能。该实验在一个玩具数据集上进行，以表明我们的方法可以很好地恢复节点之间的真实约束关系。并且还考虑使用不同的骨干预测网络来考虑各种现实世界的数据集。结果表明，借助所提出的框架可以显着减少预测误差。