Improved Constrained Generation by Bridging Pretrained Generative Models

Xiaoxuan Liang; Saeid Naderiparizi; Yunpeng Liu; Berend Zwartsenberg; Frank Wood

cs.LGMar 6, 2026

Improved Constrained Generation by Bridging Pretrained Generative Models

Authors: Xiaoxuan Liang, Saeid Naderiparizi, Yunpeng Liu, Berend Zwartsenberg, Frank Wood

Organizations: University of British Columbia · Inverted AI

Abstract

Constrained generative modeling is fundamental to applications such as robotic control and autonomous driving, where models must respect physical laws and safety-critical constraints. In real-world settings, these constraints rarely take the form of simple linear inequalities, but instead complex feasible regions that resemble road maps or other structured spatial domains. We propose a constrained generation framework that generates samples directly within such feasible regions while preserving realism. Our method fine-tunes a pretrained generative model to enforce constraints while maintaining generative fidelity. Experimentally, our method exhibits characteristics distinct from existing fine-tuning and training-free constrained baselines, revealing a new compromise between constraint satisfaction and sampling quality.

Figures & tables

Panel of Figure 1 — Figure 1: Qualitative comparison of constrained trajectory generation. We visualize samples from the same traffic intersection scene and timestep, where gray dot trajectories indicate ground-truth agent motion and orange trajectories denote model predictions. Panel 1(a) shows an unconstrained diffusion baseline, which exhibits both offroad and collision violations while executing a left turn. Panel 1(b) shows a training-free guided method, MPGD without projection ( He et al., 2024 ) that reduces violations but introduces noticeable trajectory distortion, including persistent offroad behavior and an abrupt acceleration during a right-turn maneuver toward the northeast, leading to a near-collision. Panel 1(c) shows our MBM++, which exhibits no offroad or collision violations while preserving realistic and coherent motion. All samples are generated from the same initial conditions.

Method	Collision ( $\%$ ) $\downarrow$	Boundary ( $\%$ ) $\downarrow$	r-ELBO( $\times 10^{-2}$ ) $\uparrow$	HDH ( $\times 10^{-2}$ ) $\downarrow$
Standard diffusion	$38.12\pm 0.36$	$6.29\pm 0.06$	$\mathbf{-21.7\pm 0.1}$	$28.9\pm 1.23$
MPGD w/o projection	$\mathbf{0.00\pm 0.00}$	$\mathbf{0.00\pm 0.00}$	$-27.1\pm 0.1$	$\uline{27.7\pm 0.5}$
MBM	$2.86\pm 0.04$	$1.91\pm 0.03$	$\uline{-22.2\pm 0.1}$	$28.4\pm 0.6$
DM-MBM++ (Ours)	$\uline{0.01\pm 0.00}$	$\uline{0.03\pm 0.00}$	$-22.8\pm 0.1$	$\mathbf{27.5\pm 1.3}$
Standard FM	$29.17\pm 0.31$	$3.80\pm 0.08$	$\mathbf{-37.3\pm 0.1}$	$\mathbf{28.5\pm 2.0}$
TF-Guided FM	$\mathbf{1.03\pm 0.03}$	$\mathbf{0.19\pm 0.02}$	$-52.9\pm 0.3$	$30.4\pm 1.4$

Table 1: Bouncing balls experiment results. Best results are in bold . Second best results are

\uline{\mathrm{underlined}}

Panel of Figure 2 — Figure 2: Visualization of the bouncing balls task. Panel 2(a) and panel 2(b) show two consecutive steps before and after a bounce between the balls 3 and 8. In each step, each ball is represented by a 4-dimensional vector representing its location and velocity, shown by the red arrows. Panel 2(c) presents a Pareto comparison for diffusion-based models in terms of ELBO and infraction rate. Higher ELBO and lower infraction rate indicate better performance, with preferable methods located toward the upper-left region. Our method lies near the Pareto frontier, achieving a favorable trade off between these two metrics.

Method	Collision ( $\%$ ) $\downarrow$	Offroad ( $\%$ ) $\downarrow$	$\min\text{ADE}_{6}$ $\downarrow$	$\min\text{FDE}_{6}$ $\downarrow$	$\text{MFD}_{6}$ $\uparrow$
DJINN	$0.39$	$8.12$	$0.194$	$0.459$	$2.483$
CriticSMC	$1.03$	–	$0.345$	–	$2.201$
MPGD w/o projection	$0.32$	$0.29$	$0.195$	$0.467$	$2.372$
MBM++ (Ours)	$0.27$	$0.44$	$0.180$	$0.452$	$2.352$

Table 2: Trajectory Prediction on INTERACTION ( Zhan et al., 2019 ) : DR_DEU_Merging_MT. Offroad rate and FDE are not reported for criticSMC, and we leave them blank. Our MBM++ achieves significantly lower collision and offroad rate compared to prior baselines while having the lowest

\min\mathrm{ADE}_{6}

and

\min\mathrm{FDE}_{6}

Panel of Figure 3 — Figure 3: Comparisons on DR_DEU_Merging validation dataset (merging into single-lane scenario). In panel 3(a) , the unconstrained baseline produces an offroad trajectory for the vehicle near the bottom-right, deviating beyond the right road boundary. In panel 3(b) , MPGD without projection applies an overly strong correction that pushes the same vehicle toward the left boundary, while the middle vehicle exhibits noticeable trajectory distortion. In contrast, panel 3(c) maintains all predicted trajectories within the single drivable lane, preserving realistic motion without violations.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Collision ( $\%$ ) $\downarrow$	Offroad ( $\%$ ) $\downarrow$	$\min\text{ADE}_{6}$ $\downarrow$	$\min\text{FDE}_{6}$ $\downarrow$	$\text{MFD}_{6}$ $\uparrow$
DJINN	$1.48$	$22.03$	$0.324$	$1.043$	$5.266$
CriticSMC	$0.08$	–	$0.445$	–	$3.425$
MPGD w/o projection	$1.62$	$1.78$	$0.338$	$1.000$	$4.787$
MBM++ (Ours)	$1.57$	$2.49$	$0.327$	$1.005$	$4.684$

Appendix

Table 3: Trajectory Prediction on INTERACTION ( Zhan et al., 2019 ) : DR_DEU_Roundabout_OF.

Method	Collision ( $\%$ ) $\downarrow$	Offroad ( $\%$ ) $\downarrow$	$\min\text{ADE}_{6}$ $\downarrow$	$\min\text{FDE}_{6}$ $\downarrow$	$\text{MFD}_{6}$ $\uparrow$
DJINN	$1.24$	$1.25$	$0.231$	$0.637$	$3.990$
CriticSMC	$0.09$	–	$0.448$	–	$2.871$
MPGD w/o projection	$1.13$	$0.04$	$0.229$	$0.635$	$3.882$
MBM++ (Ours)	$0.92$	$0.06$	$0.211$	$0.629$	$3.834$

Appendix

Table 4: Trajectory Prediction on INTERACTION ( Zhan et al., 2019 ) : DR_USA_Intersection_MA.

Method	Collision ( $\%$ ) $\downarrow$	Offroad ( $\%$ ) $\downarrow$	$\min\text{ADE}_{6}$ $\downarrow$	$\min\text{FDE}_{6}$ $\downarrow$	$\text{MFD}_{6}$ $\uparrow$
DJINN	$1.10$	$8.50$	$0.236$	$0.704$	$3.757$
CriticSMC	$0.07$	–	$0.444$	–	$2.974$
MPGD w/o projection	$1.13$	$0.26$	$0.233$	$0.685$	$3.594$
MBM++ (Ours)	$1.12$	$0.35$	$0.219$	$0.676$	$3.544$

Appendix

Table 5: Trajectory Prediction on INTERACTION ( Zhan et al., 2019 ) : DR_USA_Roundabout_FT.

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Deep generative models provide state-of-the-art performance across a wide array of applications, with recent studies showing increasing applicability for science and engineering. Despite a growing corpus of literature focused on the integration of physics-based constraints into the generation process, existing approaches fail to enforce strict constraint satisfaction while maintaining sample quality. In particular, training-free constrained sampling methods, while providing per-sample feasibility guarantees, introduce a fundamental mismatch between the training objective and the constrained sampling procedure, often leading to performance degradation. Identifying this training-sampling misalignment as a central limitation of current constrained generative modeling approaches, this paper proposes Constraint-Aware Flow Matching, a novel end-to-end framework that explicitly incorporates constraint projections into the training objective. By aligning the model's learned dynamics with the constrained sampling process, the proposed method mitigates distributional shift induced by projection-based corrections, enabling high-quality constrained generation. The proposed approach is evaluated on three challenging real-world benchmarks, illustrating the generality and efficacy of the method.

Abstract

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