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Wavelet Flow Matching for Multi-Scale Physics Emulation

NeurIPS 2026 · Probabilistic and scientific modeling › Molecules and proteins · approach: Flow matching generative modeling · 0 citations (OpenAlex)

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Proposes Wavelet Flow Matching (WFM), a generative PDE emulator that runs OT-CFM directly on multi-scale DWT coefficients with a U-Net, avoiding learned autoencoders.

Problem

Autoregressive emulation of multi-scale chaotic fluid PDEs needs stable long rollouts and fine-scale detail; deterministic emulators over-smooth and latent generative models need costly separately pre-trained autoencoders.

Approach

Perform optimal-transport conditional flow matching directly in a fixed, training-free wavelet space: one velocity U-Net predicts per-scale velocities for all DWT sub-bands, conditioned on current-state wavelet coefficients, PDE constants, and past frames, with the IDWT applied only at the end of sampling.

Key results

• WFM 3-scale (db4) gives the lowest spectral coherence RMSE on Rayleigh-Bénard (0.109) versus FM pixel at 0.131, and WFM 1-scale haar gives 0.241 on Active Matter vs 0.241 for FM pixel.
• WFM 1-scale achieves 1.25x, 1.52x and 1.46x inference speedup over FM pixel on TRL, RB and AM respectively.
• WFM 3-scale outperforms FM pixel in CRPS on TRL and RB systems, and is marginally worse only at early steps on AM.

Surprising findings

• The best wavelet choice varies by PDE and by number of scales (haar, db2, db4 each optimal in different settings), so no single mother wavelet dominates.
• WFM 1-scale matches FM pixel on spectral coherence for TRL and AM despite using a smaller representation.

Limitations

• Only Daubechies orthonormal wavelets were tested; symlets, coiflets and Mallat variants remain unexplored.
• The DWT implementation operates on regular grids and cannot directly handle spherical or other geometries.
• Best wavelet choice is dataset-specific and selected empirically via spectral coherence on the test setup.

Experimental setup

Datasets: The Well: Turbulent Radiative Layer 2D The Well: Rayleigh-Bénard convection The Well: Active Matter

Benchmarks: The Well TRL The Well Rayleigh-Bénard The Well Active Matter

Metrics: VRMSE CRPS spectral coherence RMSE inference speedup

Baselines: FM pixel FM fourier WDNO FNO TFNO WNO

Builds on: OT-CFM Wavelet Diffusion Neural Operator Fourier flow matching FiLM

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