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Controlled Generation with Equivariant Variational Flow Matching
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Extends Variational Flow Matching to controlled generation (end-to-end conditional training or post hoc Bayesian inference via fixed-point updates) and gives an equivariant VFM formulation for molecular generation.
Problem
Controlled generation with flow matching models under property constraints, and incorporating symmetries (rotations, translations, permutations) into flow-based molecular generation.
Approach
Casts controlled flow matching as variational inference over the posterior endpoint distribution p_t(x1|x,y); enables post hoc control of unconditional VFM models by treating a property classifier p(y|x1) as a likelihood and running a fixed-point iteration on the posterior mean, plus conditions for group-equivariant VFM.
Key results
- Controlled QM9 polarizability MAE: end-to-end G-VFM 2.05 vs EDM 2.76; Bayesian inference VI-VFM 2.25 (unconditional model only); combined 1.98.
- Joint QM9 generation: G-VFM 99.6% atom stability, 99.5% molecule stability, 98.9% validity at 100 NFE, comparable to SemlaFlow.
- Discrete QM9 generation: VFM 99.84% valid and 0.471 FCD vs Discrete FM 98.01% valid and 0.991 FCD.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace discrete flow matching with VFM | Discrete molecular generation · QM9 | FCD: 0.991 -> 0.471 | Helped |
| Replace discrete flow matching with VFM | Discrete molecular generation · ZINC250k | Valid: 96.21 -> 98.80 | Helped |
| Use G-VFM rather than EFM | Continuous molecular generation · QM9 | NLL: -115.7 -> -120.7 | Helped |
| Use G-VFM rather than SemlaFlow | Joint molecular generation · QM9 | Mol stability: 99.6 -> 99.5 | Hurt |
| Use G-VFM rather than SemlaFlow | Joint molecular generation · GEOM-Drugs | Validity: 95.2 -> 95.3 | Helped |
| Use G-VFM rather than SemlaFlow | Joint molecular generation · GEOM-Drugs | JS(E): 0.14 -> 0.18 | Hurt |
| Train G-VFM end-to-end for property control | G-VFM · QM9 conditional generation (polarizability α) | MAE: 2.76 -> 2.05 | Helped |
| Apply Bayesian inference control to an unconditional model | G-VFM · QM9 conditional generation (polarizability α) | MAE: 2.76 -> 2.25 | Helped |
| Combine end-to-end training with Bayesian inference steering | G-VFM · QM9 conditional generation (polarizability α) | MAE: 2.05 -> 1.98 | Helped |
| Use D-Flow rather than G-VFM (Both) | Conditional generation · QM9 conditional generation (polarizability α) | MAE: 1.39 -> 1.98 | Hurt |
| Remove equivariance enforcement (ablation) | G-VFM · Molecular generation tasks | Atom and molecule stability | Hurt |
Surprising findings
- Post hoc Bayesian control of an unconditional model (VI-VFM) outperforms the dedicated conditional EDM on polarizability, though D-Flow remains better (1.39 MAE) at higher inference cost.
Limitations
- Fixed-point approximation of the posterior mean is heuristic and not a full Bayesian treatment; convergence depends on log-concavity and smoothness of the classifier.
- Continuous-only QM9 and joint results lag PONITA and SemlaFlow on some metrics; G-VFM GEOM-Drugs molecule stability 96.5% vs SemlaFlow 97.7%.
- Discrete-generation results were reproduced by the authors rather than newly proposed.
Experimental setup
Datasets: QM9 ZINC250k GEOM-Drugs
Benchmarks: QM9 ZINC250k GEOM-Drugs
Metrics: valid unique FCD NLL atom stability molecule stability JS(E) NFE MAE
Baselines: EDM EquiFM GEOLDM D-Flow MoFlow EDP-GNN GraphEBM GDSS Digress Dirichlet FM Discrete FM GCDM
Builds on: Variational Flow Matching Flow Matching EDM SemlaFlow Discrete Flow Matching
Closest papers by meaning
- Variational Flow Matching for Graph Generation (NeurIPS 2024 · similarity 0.89)
- SymDrift: One-Shot Generative Modeling under Symmetries (NeurIPS 2026 · similarity 0.89)
- Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling (NeurIPS 2025 · similarity 0.88)
- Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generation (NeurIPS 2026 · similarity 0.88)
- Training Free Guided Flow-Matching with Optimal Control (ICLR 2025 · similarity 0.88)
- Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow (ICML 2025 · similarity 0.88)
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