Operator Flow Matching for Timeseries Forecasting

TL;DR

This work addresses long-horizon forecasting of high-dimensional PDE-governed dynamics by introducing TempO, a latent flow-matching model that employs a time-conditioned velocity field $v_\theta(z,t)$ within a spectral, channel-folded Fourier framework. By combining sparse conditioning with a latent encoder and a 2D Fourier-layer regressor, TempO achieves deterministic, spectrally faithful rollouts and demonstrates an upper bound on Fourier-operator approximation error, enabling parameter-efficient scaling relative to sampler-based approaches. Empirically, TempO outperforms state-of-the-art baselines across three PDE benchmarks, delivering stable 40-step forecasts with near-perfect Pearson correlations, while reducing parameter count and memory usage compared to attention-based or convolutional regressors. The results underscore the value of aligning model architecture with continuous dynamics to obtain physically consistent, efficient long-horizon forecasts, with opportunities for extension to irregular domains and broader scientific 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.

Figures (16)

• Figure 1: Prediction performance comparison for .Left: Pearson correlation across forecasted timesteps. Forty timesteps are predicted by , , and U-Net conditioned on two preceding timesteps and sampled for each proceeding step. The Pearson correlation coefficient shows significant degradation for the U-Net, oscillatory behavior and degradation for the , and consistently stable values above $0.98$ for . Right: Predicted vorticity fields. True data (a), (b), (c), and U-Net (d). At timesteps 5, 15, and 35 the and U-Net models clearly diverge, with U-Net regressing to a noisy, while TempO maintaining excellent accuracy.
• Figure 2: Spectral graphs for . Comparison of energy spectra for , a -based model, and the U-Net baseline lipmanFlowMatchingGenerative2023. The first eight Fourier modes capture 99% of the energy, with higher modes negligible. aligns closely, while the underestimates energy beyond $k=4$. The inset bar plot shows oscillating tightly around zero with small deviations, the producing larger negative deviations, and the U-Net performing markedly worse.
• Figure 3: Ablation: Fourier mode cutoffs with .
• Figure 4: Spectral Analysis of True Vorticity: Reconstruction MSE, spectral MSE, and cumulative enstrophy fraction of true Navier--Stokes vorticity data as functions of cutoff wavenumber $k_{\mathrm{cut}}$.
• Figure 5: Navier–Stokes Vorticity (Original). Ground-truth timeseries across 40 timesteps.

Theorems & Definitions (7)

• Theorem 3.1: FNO regressor constructive upper bound
• proof : Sketch of proof of Theorem \ref{['thm:FNO_upper']}
• Proposition 3.2: Transformer/UNet Sampler-based lower bound
• proof : Sketch of proof of Proposition \ref{['prop:sampler_lower']}
• Corollary 3.3: FNO vs sampler scaling
• proof : Proof of Theorem \ref{['thm:FNO_upper']}
• proof : Proof of Proposition \ref{['prop:sampler_lower']}