Coupled Diffusion-Encoder Models for Reconstruction of Flow Fields

TL;DR

This work addresses the challenge of reconstructing high-dimensional flow fields under aggressive compression while preserving their statistical and spectral properties. It introduces DiffCoder, a hybrid architecture that couples a learned encoder with a conditional diffusion-based decoder trained end-to-end, enabling generative reconstructions that respect the flow's distributional structure. On Kolmogorov flow at Re = 1000, DiffCoder outperforms a matched VAE in spectral fidelity, particularly at high wavenumbers under deep compression, while remaining competitive at moderate compression. The results suggest that diffusion-based decoding provides a promising route to compact, statistically faithful representations of complex flow fields, with potential extensions to 3D flows, latent-space forecasting, and physics-informed losses.

Abstract

Data-driven flow-field reconstruction typically relies on autoencoder architectures that compress high-dimensional states into low-dimensional latent representations. However, classical approaches such as variational autoencoders (VAEs) often struggle to preserve the higher-order statistical structure of fluid flows when subjected to strong compression. We propose DiffCoder, a coupled framework that integrates a probabilistic diffusion model with a conventional convolutional ResNet encoder and trains both components end-to-end. The encoder compresses the flow field into a latent representation, while the diffusion model learns a generative prior over reconstructions conditioned on the compressed state. This design allows DiffCoder to recover distributional and spectral properties that are not strictly required for minimizing pointwise reconstruction loss but are critical for faithfully representing statistical properties of the flow field. We evaluate DiffCoder and VAE baselines across multiple model sizes and compression ratios on a challenging dataset of Kolmogorov flow fields. Under aggressive compression, DiffCoder significantly improves the spectral accuracy while VAEs exhibit substantial degradation. Although both methods show comparable relative L2 reconstruction error, DiffCoder better preserves the underlying distributional structure of the flow. At moderate compression levels, sufficiently large VAEs remain competitive, suggesting that diffusion-based priors provide the greatest benefit when information bottlenecks are severe. These results demonstrate that the generative decoding by diffusion offers a promising path toward compact, statistically consistent representations of complex flow fields.

Figures (9)

• Figure 1: Top: DiffCoder's equivalent of a decoder is a conditional U-Net that performs iterative denoising in the reverse diffusion process. The encoder produces $\mathbf{z}$, which conditions the U-Net via concatenation to the input and optionally cross-attention. The U-Net receives timestep $t$ through shift-and-scale modulation and predicts the target (in our case velocity $\mathbf{v}$) for the reverse diffusion process. Bottom: The building blocks of DiffCoder and the VAE baseline.
• Figure 2: Attention mechanism ablation study for 8M parameter scale and encoder depth 3 (64$\times$ compression). Comparison of VAE and DiffCoder across four attention configurations: None, Encoder only, Decoder/U-Net only, and Encoder+Decoder/U-Net. Results show the impact of different attention mechanisms on normalized spectral error (lower is better).
• Figure 3: Vorticity error across model sizes and encoder depths. Both models slightly improve with increased capacity, with the VAE being the consistent winner. In deeper compressions, both models fail to achieve an acceptable performance with VAE not even reaching 40% and DiffCoder showing even worse performance.
• Figure 4: Spectral error across model sizes and encoder depths. DiffCoder demonstrates substantial advantages at encoder depth 4, where spectral content is lost the most and are the most difficult to preserve. At shallow depths, both approaches perform comparably, with DiffCoder outperforming VAE only at depth 3 for smaller model sizes, while underperforming VAE in the remainder of the experiments.
• Figure 5: High-frequency spectral error across model sizes and encoder depths. The advantage of DiffCoder is most pronounced at depth 4 where high-wavenumber content are almost if not totally lost. This metric isolates the ability to recover small-scale structures under deep compression.