Logic-Guided Vector Fields for Constrained Generative Modeling

TL;DR

This paper introduces Logic-Guided Vector Fields (LGVF), a neuro-symbolic framework that integrates logical constraints into flow-based generative models, significantly reducing constraint violations and improving sample feasibility.

Contribution

LGVF is the first method to incorporate differentiable logical constraints directly into flow matching generative models, enabling constraint-aware sampling and improved generation fidelity.

Findings

Reduces constraint violations by 59-82% across tasks.

Improves distributional fidelity in linear and ring constraints.

Emergent obstacle-avoidance behavior in generated samples.

Abstract

Neuro-symbolic systems aim to combine the expressive structure of symbolic logic with the flexibility of neural learning; yet, generative models typically lack mechanisms to enforce declarative constraints at generation time. We propose Logic-Guided Vector Fields (LGVF), a neuro-symbolic framework that injects symbolic knowledge, specified as differentiable relaxations of logical constraints, into flow matching generative models. LGVF couples two complementary mechanisms: (1) a training-time logic loss that penalizes constraint violations along continuous flow trajectories, with weights that emphasize correctness near the target distribution; and (2) an inference-time adjustment that steers sampling using constraint gradients, acting as a lightweight, logic-informed correction to the learned dynamics. We evaluate LGVF on three constrained generation case studies spanning linear, nonlinear, and multi-region feasibility constraints. Across all settings, LGVF reduces constraint violations by 59-82% compared to standard flow matching and achieves the lowest violation rates in each case. In the linear and ring settings, LGVF also improves distributional fidelity as measured by MMD, while in the multi-obstacle setting, we observe a satisfaction-fidelity trade-off, with improved feasibility but increased MMD. Beyond quantitative gains, LGVF yields constraint-aware vector fields exhibiting emergent obstacle-avoidance behavior, routing samples around forbidden regions without explicit path planning.