Logic-Guided Vector Fields (LGVF)

• LGVF is a neuro-symbolic framework that embeds differentiable logic constraints into continuous-time generative models for constraint-aware sampling.
• It combines training-time penalty terms with inference-time gradient corrections to steer sample trajectories away from constraint violations.
• Empirical evaluations in linear, nonlinear, and obstacle-avoidance settings show significant reductions in violation rates and improved distribution fidelity.

Logic-Guided Vector Fields (LGVF) are a neuro-symbolic framework for constrained generative modeling that incorporate symbolic, logic-based knowledge into continuous-time generative models, specifically flow matching architectures. LGVF injects differentiable relaxations of logical constraints into sample generation, coupling a training-time penalty for constraint violations with an inference-time corrective mechanism based on the gradients of those constraints. The method achieves substantial reductions in constraint-violation rates and can yield improved fidelity to the target distribution. LGVF establishes a scalable approach to constraint-aware sampling, evidenced by performance gains in linear, nonlinear, and obstacle-avoidance domains (Baheri, 2 Feb 2026).

1. Continuous-Time Generative Modeling and Constraints

Generative modeling by continuous-time flows frames sample generation as the solution to an ordinary differential equation (ODE) transporting points from a tractable base distribution $p_0$ to a complex target distribution $p_{\rm data}$. In the flow-matching paradigm, the dynamics are parameterized by a vector field $v_\theta(x, t)$, and the ODE

$$\dot x(t) = v_\theta(x(t), t), \quad t \in [0, 1], \quad x(0) \sim p_0$$

transports samples from $p_0$ towards $p_{\rm data}$. Standard flow matching optimizes $v_\theta$ via the conditional flow matching loss: $$\mathcal{L}_{\rm FM}(\theta) = \mathbb{E}_{t \sim \mathcal{U}[0, 1],\, x_0 \sim p_0,\, x_1 \sim p_{\rm data}} \left\| v_\theta(x_t, t) - (x_1 - x_0) \right\|^2$$ where $x_t = (1-t) x_0 + t x_1$.

Generative models of this class lack mechanisms for enforcing declarative (symbolic) constraints on $x$ during generation. LGVF addresses this by integrating logic-aware constraints directly into both training and inference phases.

2. Differentiable Relaxation of Logical Constraints

LGVF expresses symbolic constraints $\phi(x) \in \{\text{True}, \text{False}\}$ through differentiable surrogates $\ell_{\rm logic}(x) \ge 0$, where $\ell_{\rm logic}(x) = 0$ iff $\phi(x) = \text{True}$. The general form is: $\ell_{\rm logic}(x) = \Phi(h(x))$ with $h: \mathbb{R}^d \to \mathbb{R}^k$ extracting features relevant to the constraint (e.g., $h(x) = a^\top x - b$ for a half-space), and $\Phi: \mathbb{R}^k \to \mathbb{R}_{\ge 0}$ being a hinge-style relaxation such as $\Phi(u) = \max(0, -u)$.

A penalty term is added to the training objective, resulting in the total LGVF loss: $$\mathcal{L}_{\rm LGVF}(\theta) = \mathcal{L}_{\rm FM}(\theta) + \mathcal{L}_{\rm logic}(\theta)$$ where the logic loss is a time-weighted trajectory integral: $$\mathcal{L}_{\rm logic}(\theta) = \mathbb{E}_{t, x_0, x_1} \left[ \lambda(t)\, \ell_{\rm logic}(x_t) \right]$$ and the schedule $\lambda(t) = \lambda_{\max} t^\alpha$ increases toward $t=1$, where adherence to constraints becomes critical.

This approach shapes $v_\theta$ to transport mass in a way that inherently avoids constraint violation, especially near the target distribution.

3. Inference-Time Logic Adjustment

Even with robust training-time penalties, inference-time violations can occur due to the complexity of the constraint surface and model limitations. LGVF employs an inference-time "steering" correction during numerical ODE integration: $$\tilde v(x, t) = v_\theta(x, t) - \eta(t) \nabla_x \ell_{\rm logic}(x)$$ where $\eta(t) \ge 0$ is a schedule that becomes active at later times (e.g., $\eta(t) = 0$ for $t \leq 0.3$, then increasing quadratically to $\eta_{\max}$). The negative gradient $-\nabla_x \ell_{\rm logic}(x)$ points in the direction of maximal reduction in violation, nudging samples back into feasible regions without explicit path planning.

This two-stage design—combining training-time logic shaping and local inference-time steering—enables robust satisfaction of symbolic constraints across a variety of geometry classes.

4. Empirical Evaluation on Constrained Generation

LGVF was evaluated in three 2D settings: a linear half-plane, a nonlinear ring, and a multi-obstacle "forbidden disk" region. In all experiments, $v_\theta(x, t)$ was implemented as a 3-layer MLP with 128 hidden units and ReLU activations, trained using Adam for 8,000 steps (learning rate $3 \times 10^{-3}$, batch size 256), with 100 Euler steps for ODE integration at inference.

Summary of results for 2,000 samples per geometric setting:

Scenario	Violations (FM)	Violations (LGVF)	Violations (LGVF+Adj.)	MMD (FM)	MMD (LGVF)	MMD (LGVF+Adj.)
Linear half-plane ($x_1+x_2 \ge 0$)	2.20%	2.00% (9%)	0.40% (82%)	$0.89\times10^{-3}$	$0.99\times10^{-3}$	$0.29\times10^{-3}$
Nonlinear ring ($1.5 \leq \|x\| \leq 2.8$)	5.65%	3.45% (39%)	1.20% (79%)	$1.35\times10^{-3}$	$0.86\times10^{-3}$	$0.51\times10^{-3}$
Multi-obstacle avoidance	1.70%	2.50% (–47%)	0.70% (59%)	$0.40\times10^{-3}$	$0.68\times10^{-3}$	$0.82\times10^{-3}$

Percentages in parentheses denote improvement relative to baseline flow matching (FM). LGVF with inference-time adjustment ("LGVF+Adj.") consistently reduced violation rates by 59–82% across tasks. In the linear and ring settings, distributional fidelity—measured by Maximum Mean Discrepancy (MMD)—was also improved by eliminating infeasible samples. For the multi-region, obstacle scenario, improved feasibility came at the cost of a minimally higher MMD, illustrating a satisfaction–fidelity trade-off.

Empirically, LGVF induced "emergent obstacle-avoidance behavior," automatically routing generative trajectories around forbidden regions without explicit planning.

5. Implementation and Ablation Insights

Key implementation parameters included the network architecture (3-layer, 128-unit MLP), time concatenation, training setup (Adam, learning rate $3\times10^{-3}$, 8,000 steps, batch size 256), logic weight schedule $\lambda(t)=\lambda_{\max} t$ with $\lambda_{\max}\in[10,15]$, and inference schedules $\eta(t)=0$ for $t\le 0.3$ and quadratic ramp-up to $\eta_{\max} \in [0.5, 1.5]$.

Ablation studies in the linear constraint setting showed that:

• Increasing $\lambda_{\max}$ steadily reduces violation rates, reaching zero for $\lambda_{\max} \gtrsim 20$.
• Larger $\eta_{\max}$ values in the inference correction reduce violations for both FM+Adjusted and LGVF+Adjusted, but LGVF+Adjusted achieves zero errors with smaller $\eta_{\max}$, indicating complementarity between mechanisms.
• The timing of inference adjustment ($t_0$ between 0.1 and 0.5) has minimal effect, suggesting robustness of the correction mechanism.

6. Advantages, Limitations, and Future Prospects

LGVF brings hard constraint satisfaction to continuous-time flow generative models by merging training-time vector-field shaping with an inference-time gradient-based steering mechanism. Documented advantages include:

• Consistent constraint-violation reduction (59–82%).
• Emergent obstacle-avoidance without explicit path planning.
• Improved or preserved distributional fidelity (MMD), especially in convex constraint settings.
• Scalability demonstrated by near-zero violation rates up to 100 dimensions for half-space constraints.

Limitations and potential directions for further research include:

• Training-time logic shaping may be less effective on highly nonconvex or multi-region constraints; adaptive weighting or curriculum learning could mitigate this.
• Application to structured, high-dimensional data (e.g., images, molecules) invites research into learning or differentiating more complex violation measures.
• Joint learning of $\ell_{\rm logic}$ from data or symbolic programs is a promising generalization.
• Further coupling with optimal-transport techniques or more expressive vector-field architectures may yield improvements in sample quality under constraints.

LGVF exemplifies the unification of neuro-symbolic constraint satisfaction with the flexibility of continuous generative dynamics, offering a lightweight and extensible strategy for generating samples that meet complex, declarative requirements (Baheri, 2 Feb 2026).