Multivariate Long-term Time Series Forecasting with Fourier Neural Filter (2506.09174v1)

Abstract: Multivariate long-term time series forecasting has been suffering from the challenge of capturing both temporal dependencies within variables and spatial correlations across variables simultaneously. Current approaches predominantly repurpose backbones from natural language processing or computer vision (e.g., Transformers), which fail to adequately address the unique properties of time series (e.g., periodicity). The research community lacks a dedicated backbone with temporal-specific inductive biases, instead relying on domain-agnostic backbones supplemented with auxiliary techniques (e.g., signal decomposition). We introduce FNF as the backbone and DBD as the architecture to provide excellent learning capabilities and optimal learning pathways for spatio-temporal modeling, respectively. Our theoretical analysis proves that FNF unifies local time-domain and global frequency-domain information processing within a single backbone that extends naturally to spatial modeling, while information bottleneck theory demonstrates that DBD provides superior gradient flow and representation capacity compared to existing unified or sequential architectures. Our empirical evaluation across 11 public benchmark datasets spanning five domains (energy, meteorology, transportation, environment, and nature) confirms state-of-the-art performance with consistent hyperparameter settings. Notably, our approach achieves these results without any auxiliary techniques, suggesting that properly designed neural architectures can capture the inherent properties of time series, potentially transforming time series modeling in scientific and industrial applications.

• The paper introduces the Fourier Neural Filter (FNF) that unifies local time-domain and global frequency-domain processing with an efficient O(N log N) complexity.
• It proposes a Dual Branch Design (DBD) that separates temporal and spatial processing, mitigating information bottlenecks and enhancing gradient flow.
• Empirical evaluations on 11 benchmark datasets demonstrate lower MAE and state-of-the-art performance, underscoring its applicability in energy, climate, and financial forecasting.

Multivariate Long-term Time Series Forecasting with Fourier Neural Filter

The paper "Multivariate Long-term Time Series Forecasting with Fourier Neural Filter" introduces a novel methodology for multivariate long-term time series forecasting, which addresses the inherent challenges of capturing temporal dependencies and spatial correlations within the data simultaneously. The authors present the Fourier Neural Filter (FNF) as a backbone specifically designed for time series data, overcoming limitations of conventional architectures predominantly repurposed from NLP and computer vision domains, such as Transformers.

Core Contributions

The research outlines two key contributions:

1. Fourier Neural Filter (FNF):
   • The FNF is a nonlinear integral kernel operator integrating both temporal-specific inductive biases and spatial modeling capabilities.
   • It unifies local time-domain and global frequency-domain information processing within a single backbone.
   • The computational complexity of the FNF is notably efficient, scaling as $O(N \log N)$, offering significant improvements over the quadratic complexity in traditional Transformer models.
2. Dual Branch Design (DBD):
   • This architectural design is grounded in information bottleneck theory, optimizing gradient flow and representation capacity.
   • The DBD separates processing pathways for temporal and spatial patterns, allowing for independent modeling without information loss, which is prevalent in unified or sequential architectures.

Methodology

At the core of the proposed method is the FNF, which extends the standard Fourier Neural Operator (FNO) with an input-dependent kernel function. This allows adaptive modulation of information flow based on input properties, enabling selective activation of features in both time and frequency domains. Moreover, the dual-branch architecture divides the temporal and spatial modeling into two separate paths, thereby enhancing information extraction while mitigating bottleneck issues associated with sequential information processing.

The FNF and DBD are thoroughly evaluated on 11 benchmark datasets spanning diverse domains including energy, meteorology, and transportation. Across these varied datasets, the FNF consistently achieves state-of-the-art performance with a unified set of hyperparameters, which are not supplemented by auxiliary techniques such as signal decomposition.

Numerical Results

The empirical evaluation demonstrates superior performance of the proposed approach, particularly in achieving lower Mean Absolute Error (MAE) across forecast horizons compared to eight strong baseline models. These results highlight the efficacy of the proposed backbone and architecture in modeling complex temporal and spatial patterns in multivariate time series data.

Implications and Future Directions

The introduction of FNF offers significant implications for scientific and industrial applications where accurate long-term forecasts are crucial. Real-world applications such as energy grid management, climate modeling, and financial forecasting stand to benefit substantially from the improved accuracy and efficiency of this approach.

From a theoretical perspective, the integration of temporal-specific biases directly into the neural architecture presents an opportunity for rethinking time series modeling frameworks, possibly influencing subsequent research in developing specialized backbones for time series data analysis.

Future research could explore expanding the capabilities of the FNF to other time series tasks such as anomaly detection and classification, while also investigating its integration with other advanced neural architectures and frameworks. The balance between computational efficiency and accuracy remains a vital aspect to consider as the demands and scales of real-world applications continue to grow.

In conclusion, this paper provides a significant step forward in multivariate long-term time series forecasting. By combining rigorous theoretical analysis with empirical evaluations, the authors have demonstrated the capability of the Fourier Neural Filter and Dual Branch Design to advance the field beyond the limitations of existing models.