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Adaptive Flow Matching for Resolving Small-Scale Physics
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Proposes Adaptive Flow Matching (AFM), which encodes coarse inputs into a latent base distribution, injects channel-wise noise scaled by a maximum-likelihood encoder RMSE estimate, and applies flow matching to generate fine-scale physics for misaligned multiscale data.
Problem
Super-resolving small-scale physics (e.g., weather downscaling) when input and target are spatially misaligned, have different channels, and channels differ in stochasticity, where conditional diffusion/flow models and residual two-stage methods (CorrDiff) struggle.
Approach
Replace the Gaussian source of flow matching with a learned encoder output plus noise whose per-channel scale is set adaptively from the encoder's validation RMSE via a maximum-likelihood argument, trained end-to-end with a denoising (EDM-style) objective weighted by (σz/σt)^2 and an encoder regularization term.
Key results
- On CWA downscaling, AFM achieves radar CRPS 1.78 vs 1.89 for CorrDiff and 2.39 for CDM, and lower wind errors, but CorrDiff is best on temperature RMSE (0.83 vs AFM 1.00).
- On Kolmogorov Flow with tau=10, AFM reaches RMSE 1.09 vs 1.22 for CFM and 1.24 for CDM and CRPS 0.65 vs 0.65 (CDM).
- Adaptive noise scaling improves radar CRPS from 1.88 to 1.66 with a 1x1 conv encoder (Table 4).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use AFM instead of CFM | Weather downscaling, radar · CWA Taiwan | RMSE: 5.06 -> 4.90 | Helped |
| Use AFM instead of CorrDiff | Weather downscaling, temperature · CWA Taiwan | RMSE: 0.83 -> 1.00 | Hurt |
| Use AFM instead of CFM | Weather downscaling, eastward wind · CWA Taiwan | CRPS: 0.82 -> 0.80 | Helped |
| Use AFM instead of CFM | Weather downscaling, northward wind · CWA Taiwan | CRPS: 0.90 -> 0.88 | Helped |
| Use AFM instead of the strongest baseline | Kolmogorov Flow · Kolmogorov Flow τ=3 | RMSE: 0.98 -> 0.73 | Helped |
| Use AFM instead of the strongest baseline | Kolmogorov Flow · Kolmogorov Flow τ=5 | RMSE: 0.94 -> 0.76 | Helped |
| Use AFM instead of the strongest baseline | Kolmogorov Flow · Kolmogorov Flow τ=10 | CRPS: 0.65 -> 0.65 | No clear effect |
| Use adaptive σz rather than fixed σz (ablation) | AFM with 1×1 Conv. encoder · CWA 112×112 Radar | RMSE: 5.01 -> 4.82 | Helped |
| Use adaptive σz rather than fixed σz (ablation) | AFM with UNet encoder · CWA 112×112 Radar | RMSE: 5.06 -> 4.95 | Helped |
| Set λ=0.25 rather than λ=0.00 (ablation) | AFM with UNet encoder · CWA 112×112 Radar | RMSE: 5.06 -> 4.95 | Helped |
| Set λ=0.25 rather than λ=0.00 (ablation) | AFM with 1×1 Conv. encoder · CWA 112×112 Radar | RMSE: 4.82 -> 5.10 | Hurt |
Surprising findings
- CDM appears the best calibrated on Kolmogorov Flow, yet AFM still has superior overall skill.
- The simpler 1x1 convolution encoder (60 parameters) performs best with no regularization (lambda=0).
Limitations
- AFM does not achieve perfect ensemble calibration (SSR not equal to 1).
- CorrDiff outperforms AFM on deterministic temperature RMSE.
- Requires paired data; extension to unpaired misaligned datasets left to future work.
Experimental setup
Datasets: ERA5 CWA Taiwan observations Multiscale Kolmogorov Flow (synthetic)
Benchmarks: CWA Taiwan downscaling 2021 test set Multiscale Kolmogorov Flow
Metrics: RMSE CRPS MAE SSR power spectra
Baselines: UNet regression conditional diffusion model conditional flow matching CorrDiff
Builds on: Flow Matching EDM CorrDiff
Closest papers by meaning
- Heavy-Tailed Flow Matching via Random Clocks (NeurIPS 2026 · similarity 0.88)
- On Fitting Flow Models with Large Sinkhorn Couplings (NeurIPS 2026 · similarity 0.87)
- PrismFlow: Residual Dynamics for Flow Matching in Time-Series Generation (NeurIPS 2026 · similarity 0.87)
- Coreset-Induced Conditional Velocity Flow Matching (NeurIPS 2026 · similarity 0.86)
- Gaussian Mixture Flow Matching Models (ICML 2025 · similarity 0.86)
- Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers (ICML 2025 · similarity 0.86)
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