EDGE-VARYING FOURIER GRAPH NETWORKS FOR MULTIVARIATE TIME SERIES FORECASTING Anonymous authors Paper under double-blind review

Abstract

The key to multivariate time series (MTS) analysis and forecasting is to disclose the underlying couplings between variables that drive the co-movements. Considerable recent successful MTS methods are built with graph neural networks (GNNs) due to their essential capacity for relational modeling. However, previous work often used a static graph structure of time-series variables for modeling MTS, but failed to capture their ever-changing correlations over time. In this paper, we build a fully-connected supra-graph, representing non-static correlations between any two variables at any two timestamps, to capture high-resolution spatial-temporal dependencies. Whereas, conducting graph convolutions on such a supra-graph heavily increases computational complexity. As a result, we propose the novel Edge-Varying Fourier Graph Networks (EV-FGN), which reformulates the graph convolutions in the frequency domain with high efficiency and scale-free parameters, and applies edge varying graph filters to capture the time-varying variable dependencies. Extensive experiments show that EV-FGN outperforms state-of-the-art methods on seven real-world MTS datasets.



Despite the success of GNNs on MTS forecasting, three practical challenges are eagerly demanded to address: 1) the dependencies between each pair of time series variables are generally non-static, which demands full dependencies modeling with all possible lags; 2) a high-efficiency dense graph learning method is demanded to replace the high-cost operators of graph convolutions and attention (with quadratic time complexity of the graph size); 3) the graph over MTS varies with different temporal interactions, which demands an efficient dynamic structure encoding method. In this paper, different from most GNN-based methods that construct graphs to model the spatial correlation between variables (i.e. nodes) Bai et al. (2020); Wu et al. (2019; 2020) , we attempt to build a supra-graph that sheds light on the "high-resolution" correlations between any two variables at any two timestamps (i.e., fine-grained spatial-temporal dependencies), and largely enhances the expressiveness on non-static spatial-temporal dependencies. Obviously, the supra-graph will heavily increase the computational complexity of GNN-based model, then a high-efficiency learning method is required to reduce the cost for model training. Inspired by Fourier Neural Operator (FNO) Li et al. ( 2021), we reformulate the graph convolution (time domain) to much lower-complexity matrix multiplication in the frequency domain by leveraging an efficient and newly-defined Fourier Graph Shift Operator (FGSO). In addition, it is necessary to consider the multiple iterations (layers) of graph convolutions to expand receptive neighbors and mix the diffusion 1



(MTS) forecasting is a key ingredient in many real-world scenarios, including weather forecasting Zheng et al. (2015), decision making Borovykh et al. (2017), traffic forecasting Yu et al. (2018a); Bai et al. (2020), COVID-19 prediction Cao et al. (2020); Chen et al. (2022), etc. Recently, deep neural networks, such as long short-term memory (LSTM) Hochreiter & Schmidhuber (1997), convolutional neural network (CNN) Borovykh et al. (2017), Transformer Vaswani et al. (2017), have dominated MTS modeling. In particular, graph neural networks (GNNs) have demonstrated promising performance on MTS forecasting with their essential capability to capture the complex couplings between time-series variables. Some studies enable to adaptively learn the graph for MTS forecasting even without an explicit graph structure, e.g., by node similarity Mateos et al. (2019); Bai et al. (2020); Wu et al. (2019) and/or self-attention mechanism Cao et al. (2020).

