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Training Free Guided Flow-Matching with Optimal Control
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Proposes OC-Flow, a training-free guided flow-matching framework that formulates reward-guided ODE generation as optimal control solved with an extended method of successive approximations, extended to SO(3) for peptide design.
Problem
Training-free guidance of pre-trained ODE-based flow models toward a target reward while staying close to the prior, with little theoretical grounding and no treatment of non-Euclidean manifolds such as SO(3).
Approach
Casts guided flow generation as an optimal control problem with additive (Euclidean) or left-invariant (SO(3)) control and a running cost; solves it with Extended MSA using co-state gradients computed by vector-Jacobian products, and shows D-Flow and FlowGrad are special cases.
Key results
- On QM9 conditional generation, OC-Flow's MAE is lower than D-Flow and FlowGrad across properties (e.g., alpha MAE 1.383 with gamma=0.01).
- In peptide design, OC-Flow(trans+rot) reaches MadraX -0.263 and stability -48.853 versus PepFlow's -0.195 and -45.660, with IMP 15.0% vs 14.3%.
- Image sampling takes 216s versus about 15 minutes for D-Flow, per the authors.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| guide both translations and rotations with OC-Flow | PepFlow w/Bb · Peptide design test set | MadraX energy: -0.195 -> -0.263 | Helped |
| guide both translations and rotations with OC-Flow | PepFlow w/Bb · Peptide design test set | RMSD (Å): 1.645 -> 2.127 | Hurt |
| guide both translations and rotations with OC-Flow | PepFlow w/Bb · Peptide design test set | SSR: 0.794 -> 0.797 | Helped |
| guide both translations and rotations with OC-Flow | PepFlow w/Bb · Peptide design test set | BSR: 0.874 -> 0.869 | Hurt |
| guide both translations and rotations with OC-Flow | PepFlow w/Bb · Peptide design test set | Stability: -45.660 -> -48.853 | Helped |
| guide both translations and rotations with OC-Flow | PepFlow w/Bb · Peptide design test set | Affinity: -26.538 -> -27.468 | Helped |
| guide both translations and rotations with OC-Flow | PepFlow w/Bb · Peptide design test set | Diversity: 0.310 -> 0.338 | Helped |
| guide both translations and rotations with OC-Flow | PepFlow w/Bb · Peptide design test set | IMP (%): 14.3 -> 15.0 | Helped |
| optimize translations only instead of translations and rotations (ablation) | OC-Flow · Peptide design test set | IMP (%): 15.0 -> 14.4 | No clear effect |
| optimize rotations only instead of translations and rotations (ablation) | OC-Flow · Peptide design test set | MadraX energy: -0.263 -> -0.221 | Hurt |
| increase regularization strength from γ=0.01 to γ=10 (ablation) | OC-Flow · QM9 α | MAE: 1.383 / 1.557 | Hurt |
| increase regularization strength from γ=0.01 to γ=10 (ablation) | OC-Flow · QM9 α | ASP: 94.8 / 96.0 | Helped |
Surprising findings
- Increasing the regularization strength gamma improves ASP, MSP and VUP at the cost of higher MAE, showing a trade-off between reward optimization and prior fidelity.
Limitations
- Backprop-through-ODE methods are computationally more expensive than posterior sampling approaches.
- Restricted to additive (and left-invariant) control terms.
- Convergence analysis relies on Lipschitz assumptions that are only assumed, not verified, for deep models.
Experimental setup
Datasets: CelebA-HQ QM9 PepFlow peptide-protein complex test set (162 complexes)
Benchmarks: CelebA-HQ text-guided manipulation QM9 conditional generation peptide design test set
Metrics: LPIPS ID CLIP score MAE ASP MSP VUP MadraX energy RMSD SSR
Baselines: StyleCLIP FlowGrad D-Flow PepFlow (unconditional)
Builds on: D-Flow FlowGrad Extended Method of Successive Approximations Rectified Flow PepFlow
Closest papers by meaning
- Greed is Good: A Unifying Perspective on Guided Generation (NeurIPS 2025 · similarity 0.91)
- On the Guidance of Flow Matching (ICML 2025 · similarity 0.90)
- Flow Density Control: Generative Optimization Beyond Entropy-Regularized Fine-Tuning (NeurIPS 2025 · similarity 0.90)
- Optimal Flow Matching: Learning Straight Trajectories in Just One Step (NeurIPS 2024 · similarity 0.89)
- D-Flow: Differentiating through Flows for Controlled Generation (ICML 2024 · similarity 0.88)
- Path-Guided Flow Matching for Dataset Distillation (NeurIPS 2026 · similarity 0.88)
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