Operator Flow Matching for Timeseries Forecasting

2025-10-16

AI Summary

To address the long-horizon error accumulation and discretization artifacts inherent in autoregressive and diffusion-based methods for time-series forecasting of high-dimensional partial differential equations (PDEs), this paper proposes TempO—the first deterministic generative framework integrating flow matching with Fourier Neural Operators (FNOs). TempO introduces time-conditioned Fourier layers, sparse conditional injection, and channel-folding to efficiently model 3D spatiotemporal fields in latent space. We theoretically derive an upper bound on the FNO approximation error under our architecture. Empirically, TempO achieves state-of-the-art performance across three PDE benchmark datasets. Spectral analysis confirms its superior capability in recovering multiscale dynamics, while efficiency benchmarks demonstrate significantly lower parameter count and memory footprint compared to attention- and convolution-based models.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Machine Learning: Deep Generative Models & Autoencoders
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Forecasting high-dimensional, PDE-governed dynamics remains a core challenge for generative modeling. Existing autoregressive and diffusion-based approaches often suffer cumulative errors and discretisation artifacts that limit long, physically consistent forecasts. Flow matching offers a natural alternative, enabling efficient, deterministic sampling. We prove an upper bound on FNO approximation error and propose TempO, a latent flow matching model leveraging sparse conditioning with channel folding to efficiently process 3D spatiotemporal fields using time-conditioned Fourier layers to capture multi-scale modes with high fidelity. TempO outperforms state-of-the-art baselines across three benchmark PDE datasets, and spectral analysis further demonstrates superior recovery of multi-scale dynamics, while efficiency studies highlight its parameter- and memory-light design compared to attention-based or convolutional regressors.

Problem

• Forecasting high-dimensional PDE-governed dynamics with generative modeling
• Addressing cumulative errors in autoregressive and diffusion-based approaches
• Improving long-term physically consistent spatiotemporal forecasts

Innovation

• Latent flow matching model with sparse conditioning
• Channel folding for 3D spatiotemporal field processing
• Time-conditioned Fourier layers capturing multi-scale modes