Verifier-Constrained Flow Expansion for Discovery Beyond the Data

Riccardo De Santi; Kimon Protopapas; Ya-Ping Hsieh; Andreas Krause

cs.LGFeb 17, 2026

Verifier-Constrained Flow Expansion for Discovery Beyond the Data

Authors: Riccardo De Santi, Kimon Protopapas, Ya-Ping Hsieh, Andreas Krause

Organizations: ETH Zürich · ETH AI Center

Abstract

Flow and diffusion models are typically pre-trained on limited available data (e.g., molecular samples), covering only a fraction of the valid design space (e.g., the full molecular space). As a consequence, they tend to generate samples from only a narrow portion of the feasible domain. This is a fundamental limitation for scientific discovery applications, where one typically aims to sample valid designs beyond the available data distribution. To this end, we address the challenge of leveraging access to a verifier (e.g., an atomic bonds checker), to adapt a pre-trained flow model so that its induced density expands beyond regions of high data availability, while preserving samples validity. We introduce formal notions of strong and weak verifiers and propose algorithmic frameworks for global and local flow expansion via probability-space optimization. Then, we present Flow Expander (FE), a scalable mirror descent scheme that provably tackles both problems by verifier-constrained entropy maximization over the flow process noised state space. Next, we provide a thorough theoretical analysis of the proposed method, and state convergence guarantees under both idealized and general assumptions. Ultimately, we empirically evaluate our method on both illustrative, yet visually interpretable settings, and on a molecular design task showcasing the ability of FE to expand a pre-trained flow model increasing conformer diversity while preserving validity.

Figures & tables

Figure 1: Limited coverage of the valid design space leads to generating sub-optimal samples for downstream optimization tasks.

Panel of Figure 2 — Figure 2: ( 2(a) ) Pre-trained and globally expanded flow model inducing densities $p^{pre}_{1}$ and optimal density $p_{1}^{*}$ . ( 2(b) ) Valid design space $\Omega$ , strong and weak verifiers $\Omega_{v_{i}}$ , $i\in[3]$ , and their compositions.

\tilde{\pi}^{k}\leftarrow\textsc{{FineTuningSolver}}(\pi^{k-1},\nabla_{x_{t}}\delta\mathcal{G}_{t},\lambda_{t},\gamma_{k})

(12)

Algorithm 1 ExpandThenProject

\displaystyle\nabla_{x_{t}}\delta\mathcal{G}_{t}(p_{t}^{k-1})

\displaystyle=\left\{\begin{aligned} &-s^{\pi^{k-1}}_{t}&&\quad\text{{G}lobal {FE} ({{G-FE}})}\\ &-s^{\pi}_{t}-\alpha_{t}\left(s^{\pi}_{t}-s^{\textrm{pre}}_{t}\right)&&\quad\text{{L}ocal {FE} ({{L-FE}})}\end{aligned}\right.

(14)

Algorithm 2 F low E xpander ( FE )

\nabla_{x_{t}}\delta\mathcal{G}_{t}(p_{t}^{k-1})=-s^{\pi}_{t}-\alpha_{t}\left(s^{\pi}_{t}-s^{\textrm{pre}}_{t}\right)

(20)

Algorithm 3 N oised S pace E xploration ( NSE )

Panel of Figure 3 — Figure 3: (top) G lobal FE ( G-FE ) expands the pre-trained flow model $\pi^{pre}$ ( 3(a) ) into $\pi^{*}$ (violet, 3(b) ), increasing coverage (i.e., entropy), while preserving validity (i.e., red ellipse interior). Compared with the unconstrained exploration S-MEME method, and constrained generation ( CONSTR ), G lobal FE ( G-FE ) shows best-of-both-worlds behaviour: achieving near-optimal entropy and validity (Fig. 3(d) ).

Panel of Figure 5 — Figure 5: (top) L-FE (yellow, 5(c) ) expands the pre-trained flow model $\pi^{pre}$ (green, 5(a) ) over promising yet verifier-filtered modes, while FDC (blue, 5(b) ) expands $\pi^{pre}$ over all plausible modes leading to increased density in invalid regions (left mode in Fig. 5(b) ). (bottom) FE increases visual ( 5(d) ), and quantitative diversity ( 5(f) ), while preserving higher validity than FDC ( 5(e) - 5(g) )

Panel of Figure 6 — Figure 6: ( 6(a) ) L-FE (violet) achieves higher diversity and validity than FDC on GEOM-Drugs. ( 6(b) ) NSE , the unconstrained variant of L-FE , exhibits superior performance compared with FDC , a state-of-the-art diffusion-based unconstrained exploration method. ( 6(c) ) Computational cost comparison of L-FE , FDC , and NSE . ( 6(d) ) Representative drug-like molecule generated via L-FE .

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

X_{t+h}=X_{t}+h\Big(2u_{\theta}^{\text{finetune}}(X_{t},t)-\tfrac{\dot{\omega}_{t}}{\omega_{t}}X_{t}\Big)+\sqrt{h}\,\sigma(t)\,\varepsilon_{t},\quad\varepsilon_{t}\sim\mathcal{N}(0,I),\quad X_{0}\sim\mathcal{N}(0,I).

(51)

Appendix

Algorithm 4 Adjoint Matching for fine-tuning Flow Matching models ( FineTuningSolver )

Figure 7: Comparison of different parametrization of L-FE , G-FE , NSE , FDC , for $K=3$ . We compute Conformer Vendi and validity over 5000 samples, and average results over 8 runs for each parameterization with 8 different seeds.

$\beta$	$\tilde{\gamma}_{0}$	$\eta_{0}$	Mean Validity (95% CI)	Mean Conformer Vendi (95% CI)
0.3	0.0002	5.0	0.89 (0.87 $\pm$ 0.91)	481.35 (460.24 $\pm$ 500.42)
0.6	0.0005	5.0	0.85 (0.79 $\pm$ 0.89)	500.11 (460.72 $\pm$ 546.91)
0.4	0.0005	5.0	0.83 (0.76 $\pm$ 0.89)	519.05 (454.45 $\pm$ 585.99)
0.1	0.0004	5.0	0.82 (0.70 $\pm$ 0.90)	529.47 (465.92 $\pm$ 629.29)
0.2	0.0004	5.0	0.74 (0.53 $\pm$ 0.88)	531.90 (452.14 $\pm$ 633.92)
0.2	0.0005	5.0	0.70 (0.53 $\pm$ 0.86)	563.75 (472.19 $\pm$ 683.79)

Appendix

Table 1: Values for L-FE reported in Fig 7 with 95% confidence intervals

$\tilde{\gamma}_{0}$	$\eta_{0}$	Mean Validity (95% CI)	Mean Conformer Vendi (95% CI)
0.0002	0.5	0.76 (0.69 $\pm$ 0.83)	485.65 (442.52 $\pm$ 524.71)
0.0005	5.0	0.69 (0.46 $\pm$ 0.86)	890.19 (434.36 $\pm$ 1731.52)
0.0004	0.5	0.57 (0.33 $\pm$ 0.79)	1004.79 (531.74 $\pm$ 1790.78)
0.0004	1.0	0.52 (0.27 $\pm$ 0.77)	1125.94 (521.23 $\pm$ 2182.77)
0.0004	0.1	0.48 (0.22 $\pm$ 0.72)	1676.67 (533.47 $\pm$ 2888.04)
0.0005	1.0	0.46 (0.22 $\pm$ 0.71)	1706.53 (664.86 $\pm$ 3100.05)

Appendix

Table 2: Values for G-FE reported in Fig 7 with 95% confidence intervals

$\beta$	$\tilde{\gamma}_{0}$	Mean Validity (95% CI)	Mean Conformer Vendi (95% CI)
0.0	0.0002	0.74 (0.68 $\pm$ 0.81)	518.92 (505.70 $\pm$ 531.90)
0.4	0.0004	0.68 (0.58 $\pm$ 0.77)	529.85 (484.34 $\pm$ 580.24)
0.3	0.0004	0.68 (0.55 $\pm$ 0.79)	537.83 (491.25 $\pm$ 588.98)
0.1	0.0004	0.60 (0.42 $\pm$ 0.77)	639.04 (501.23 $\pm$ 857.00)
0.2	0.0004	0.57 (0.37 $\pm$ 0.74)	902.23 (534.29 $\pm$ 1559.95)
0.1	0.0005	0.48 (0.25 $\pm$ 0.69)	1678.29 (669.60 $\pm$ 3024.30)

Appendix

Table 3: Values for NSE reported in Fig 7 with 95% confidence intervals

$\beta$	$\tilde{\gamma}_{0}$	Mean Validity (95% CI)	Mean Conformer Vendi (95% CI)
0.4	0.0001	0.71 (0.67 $\pm$ 0.74)	484.34 (461.47 $\pm$ 506.75)
0.5	0.0002	0.71 (0.63 $\pm$ 0.77)	496.06 (467.29 $\pm$ 523.97)
0.7	0.0002	0.70 (0.64 $\pm$ 0.75)	502.49 (483.86 $\pm$ 526.78)
0.5	0.0001	0.69 (0.65 $\pm$ 0.74)	505.84 (468.66 $\pm$ 538.20)
0.2	0.0003	0.67 (0.58 $\pm$ 0.77)	539.29 (504.70 $\pm$ 585.65)
0.9	0.001	0.56 (0.49 $\pm$ 0.62)	601.06 (531.19 $\pm$ 699.07)

Appendix

Table 4: Values for FDC reported in Fig 7 with 95% confidence intervals

Figure 8: Illustrating the result of isolating the EXPAND step ( NSE ) and the PROJECT step

Figure 9: L-FE compared to NSE and FDC with inference-time filtering for different parameterizations. We note that L-FE maintains competitive performance with both methods despite having no filter applied to the output samples, and NSE still outperforms FDC when both are corrected with an inference-time filter.

Flow-Based Generative Modeling

Flow-based generative modeling learns probability transports from simple noise to data for fast sampling, density estimation, and controllable generation.

See all 552 papers

Active Flow Expansion for Out-of-Distribution Discovery: from Theory to Molecules
Jun 7, 2026Riccardo De Santi, Bruce Lee, Cristian Perez Jensen +6Synthetic Data Pretraining Normalizing Flows
ETH Zurich ETH AI Center · University of Pennsylvania · Caltech FutureHouse
Constrained Flow Optimization via Sequential Fine Tuning for Molecular Design
May 28, 2026Sven Gutjahr, Riccardo De Santi, Luca Schaufelberger +2Constrained Generative Modeling Molecular Optimization
Department of Computer Science, ETH Zurich · ETH AI Center · NCCR Catalysis, Switzerland +1
Property-Guided Molecular Generation and Optimization via Latent Flows
Mar 27, 2026Alexander Arjun Lobo, Urvi Awasthi, Leonid ZhukovMolecular Generation Molecular Optimization
BCG X AI Science Institute

Jun 7, 2026J/K move · Enter open · S save

Active Flow Expansion for Out-of-Distribution Discovery: from Theory to Molecules

Riccardo De Santi, Bruce Lee, Cristian Perez Jensen, Kimon Protopapas+5

ETH Zurich ETH AI Center · University of Pennsylvania · Caltech FutureHouse

Synthetic Data Pretraining Normalizing Flows Generative Modeling Molecular Generation Flow-Based Generative Modeling

Open paper AskPDF arXiv

Standard flow and diffusion pre-training matches the distribution of available data (e.g., molecules), which often covers only a small fraction of the valid design space. In generative discovery, however, one aims to sample valid new-to-nature designs, assigned negligible probability under, and thus inaccessible to, standard models fitted to the observed data. To overcome this limitation, we depart from data distribution matching and view a generative model through its generable set: the region it covers with non-negligible probability. This allows to introduce a new learning principle for out-of-distribution flow modeling: enlarging a model's generable set to increase coverage of the valid design space. We propose Active Flow Expansion (ActFlow), a continued pre-training method that employs verifier feedback to expand a pre-trained model over new valid regions by iteratively adapting to synthetic data generated through active exploration in the learned flow representation. Theoretically, we establish to our knowledge first-of-their-kind statistical learning guarantees for out-of-distribution flow modeling, analyzing generable set expansion as a local-to-global reachability process over a learned representation. Empirically, we assess ActFlow with suitable out-of-distribution generative modeling metrics across small organic molecules, mid-sized drug-like molecules, therapeutic peptides, and protein sequence design tasks. Results show that ActFlow expands valid coverage far beyond the region modeled by the initial pre-trained model, significantly outperforming widely adopted synthetic flow pre-training methods.

Abstract

Flow-Based Generative Modeling

Explore similar work

Active Flow Expansion for Out-of-Distribution Discovery: from Theory to Molecules

Constrained Flow Optimization via Sequential Fine Tuning for Molecular Design

Property-Guided Molecular Generation and Optimization via Latent Flows