FreqFlow: Long-term forecasting using lightweight flow matching

TL;DR

FreqFlow addresses deterministic long-term MTS forecasting under computational constraints by moving forecasting to the spectral domain and employing conditional flow matching. It introduces a frequency-domain framework with a single complex-valued linear layer to model amplitude and phase shifts, realized through an ODE $\frac{d x_t}{d t} = u_\theta(x_t, t)$. The approach yields a compact model (around $89k$ parameters) that achieves state-of-the-art or competitive accuracy on traffic datasets while enabling fast, single-pass inference and interpretable decomposition into trend/seasonality/residuals. This work offers a scalable solution for real-world deployment on resource-constrained systems and highlights the practicality of spectral-flow modeling for multivariate spatio-temporal data.

Abstract

Multivariate time-series (MTS) forecasting is fundamental to applications ranging from urban mobility and resource management to climate modeling. While recent generative models based on denoising diffusion have advanced state-of-the-art performance in capturing complex data distributions, they suffer from significant computational overhead due to iterative stochastic sampling procedures that limit real-time deployment. Moreover, these models can be brittle when handling high-dimensional, non-stationary, and multi-scale periodic patterns characteristic of real-world sensor networks. We introduce FreqFlow, a novel framework that leverages conditional flow matching in the frequency domain for deterministic MTS forecasting. Unlike conventional approaches that operate in the time domain, FreqFlow transforms the forecasting problem into the spectral domain, where it learns to model amplitude and phase shifts through a single complex-valued linear layer. This frequency-domain formulation enables the model to efficiently capture temporal dynamics via complex multiplication, corresponding to scaling and temporal translations. The resulting architecture is exceptionally lightweight with only 89k parameters - an order of magnitude smaller than competing diffusion-based models-while enabling single-pass deterministic sampling through ordinary differential equation (ODE) integration. Our approach decomposes MTS signals into trend, seasonal, and residual components, with the flow matching mechanism specifically designed for residual learning to enhance long-term forecasting accuracy. Extensive experiments on real-world traffic speed, volume, and flow datasets demonstrate that FreqFlow achieves state-of-the-art forecasting performance, on average 7\% RMSE improvements, while being significantly faster and more parameter-efficient than existing methods

Figures (6)

• Figure 1: Problem setup and our proposed method, which is flow matching with a lightweight network in the frequency space. The underlying graph depicts Brussels’s road network in Belgium; the time-series signals are for the period 08-08-2025,14:55:00 for three days. Note that in the Node B, although the traffic pattern is highly correlated to the adjacent nodes, the traffic volume is significantly less.
• Figure 2: The FrèqFlow Pipeline.
• Figure 3: Computational cost and prediction accuracy trade-off plot. These results are benchmarked on the Brussels dataset. Each baseline is annotated with the number of parameters.
• Figure 4: Loss function coefficient hyperparameter optimization results.
• Figure 5: Sensitivity analysis of hyperparameters.