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On the Guidance of Flow Matching
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Proposes a general framework for energy guidance of flow matching with arbitrary source distributions and couplings, yielding MC-based training-free, training-based, and approximate guidance methods that generalize diffusion guidance.
Problem
Guidance (energy-guided or posterior sampling) of flow matching models is underexplored for general flow matching with non-Gaussian source distributions and dependent couplings, where diffusion guidance does not apply directly.
Approach
Derives a general guidance vector field that reweights the conditional paths by exp(-J) and the reverse-coupling ratio, then gives an unbiased importance-sampling Monte Carlo estimator (training-free), training losses for a learned guidance network (guidance matching), and approximate Taylor/Gaussian-based guidances (cov-A, cov-G, sim-MC, sim-A) that recover DPS, LGD, and PiGDM as special cases.
Key results
- On D4RL Locomotion, g^MC achieves the best average normalized score of 81.0 with OT-CFM, versus 77.7 for g^cov-G and 55.9 without guidance.
- On CelebA-HQ image inverse problems, ΠGDM is generally best (best in 8 of 12 metrics), while g^MC performs poorly in high dimensions (e.g., FID ~22.8 on inpainting).
- On 2D synthetic datasets, g^MC and g_phi closely match ground-truth weighted distributions, while exact diffusion guidance (CEG) distorts them.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Monte Carlo guidance instead of unguided sampling | OT-CFM · D4RL Locomotion average | normalized score: 55.9 -> 81.0 | Helped |
| Use Monte Carlo guidance instead of unguided sampling | CFM · D4RL Locomotion average | normalized score: 54.3 -> 79.4 | Helped |
| Use Jacobian-based covariance guidance instead of manually scheduled covariance guidance | OT-CFM and CFM · D4RL Locomotion average | normalized score: 68.4 -> 77.7; 69.2 -> 75.8 | Helped |
| Use training-based guidance instead of the best-performing method | OT-CFM and CFM · D4RL Locomotion average | normalized score: 10.4 | Hurt |
| Use g^MC instead of g^cov-G | OT-CFM and CFM · D4RL Locomotion average | normalized score: 3.5 | Helped |
| Use g^MC rather than g_phi or g^MC with a smaller sample budget | Non-Gaussian synthetic flow matching · 2D synthetic datasets | Wasserstein-2 distance to the guided target distribution | Helped |
| Use diffusion contrastive energy guidance rather than general flow-matching guidance | Non-Gaussian flow matching · 2D synthetic datasets | match to the guided target distribution | Hurt |
| Use g^MC rather than the other tested methods | OT-CFM and CFM · CelebA-HQ image inverse problems | FID | Hurt |
| Use PiGDM rather than the other tested methods | OT-CFM and CFM · CelebA-HQ image inverse problems | best-in-metric count: 8 out of 12 metrics | Helped |
| Use manually scheduled covariance guidance rather than the other tested methods | OT-CFM and CFM · CelebA-HQ image inverse problems | relative metric ranking: 11 out of 12 metrics | Hurt |
Surprising findings
- The Monte Carlo training-free guidance outperforms gradient-based guidance on offline RL planning, yet performs poorly on high-dimensional image inverse problems.
- Exact diffusion guidance (CEG) performs poorly on non-Gaussian-source flow matching, since its exactness only holds under uncoupled affine Gaussian paths.
Limitations
- The method assumes the reverse coupling ratio P is approximately 1, which fails for strong couplings where guidance direction becomes incorrect.
- g^MC has high variance and scales poorly to high-dimensional data such as images.
- Learned guidance g_phi trains two networks jointly and is unstable, performing weaker on offline RL.
Experimental setup
Datasets: D4RL Locomotion CelebA-HQ 2D synthetic datasets
Benchmarks: D4RL Locomotion CelebA-HQ inpainting CelebA-HQ super-resolution CelebA-HQ Gaussian deblurring
Metrics: normalized score FID LPIPS PSNR SSIM Wasserstein-2 distance
Baselines: Diffuser BC ΠGDM contrastive energy guidance (CEG) OT-ODE
Builds on: conditional flow matching diffusion posterior sampling (DPS) ΠGDM loss-guided diffusion (LGD) Diffuser
Closest papers by meaning
- Value Gradient Guidance for Flow Matching Alignment (NeurIPS 2025 · similarity 0.92)
- Training Free Guided Flow-Matching with Optimal Control (ICLR 2025 · similarity 0.90)
- Optimal Flow Matching: Learning Straight Trajectories in Just One Step (NeurIPS 2024 · similarity 0.90)
- Greed is Good: A Unifying Perspective on Guided Generation (NeurIPS 2025 · similarity 0.89)
- Energy-Weighted Flow Matching for Offline Reinforcement Learning (ICLR 2025 · similarity 0.89)
- On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity (NeurIPS 2025 · similarity 0.89)
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