physics-constrained sampling

This work addresses the misalignment between training objectives and sampling requirements in existing constrained generative models, which often struggle to simultaneously ensure strict constraint satisfaction and high sample quality. To resolve this issue, the authors propose an end-to-end constraint-aware flow matching framework that explicitly embeds a constraint projection operator into the training objective. This design aligns the generative dynamics with the constrained sampling process, thereby avoiding the distributional shift typically induced by post-hoc projection. The method enables, for the first time, fully differentiable end-to-end optimization across both training and constrained sampling stages. Extensive experiments on three real-world benchmark tasks demonstrate the approach’s generality and effectiveness, achieving significant improvements in both sample quality and constraint satisfaction rates.

Generative models often produce infeasible samples by ignoring physical or task-specific hard constraints. Existing projection-based methods distort the underlying data distribution, while multi-stage deferred projection introduces error accumulation and implementation complexity. This paper proposes Chance-constrained Flow Matching (CCFM), a training-free method that— for the first time—incorporates chance constraints directly into the flow matching framework. CCFM enforces constraints during stochastic differential equation (SDE) sampling by operating on intermediate noise states. Theoretically, it is equivalent to a single optimal projection onto the feasible set for clean samples, thereby avoiding both iterative projection distortion and multi-stage error propagation. By unifying flow matching, SDE-based sampling, and stochastic optimization, CCFM jointly ensures constraint satisfaction and high-fidelity generation. Experiments on partial differential equation solving and molecular docking demonstrate that CCFM significantly outperforms state-of-the-art methods, achieving simultaneous improvements in feasibility and sample quality.

This work addresses the challenge in constrained generative sampling where ill-timed constraint enforcement yields samples that satisfy hard constraints yet deviate from desired dynamics. The authors formulate constraint application as a correction scheduling problem along generative trajectories and propose a state-dependent adaptive correction mechanism that dynamically allocates projection resources based on per-step constraint violations and geometric mismatch signals. This approach unifies terminal and stepwise projection strategies and, for the first time, treats the timing of constraint enforcement as a core design variable. Experiments demonstrate that under an identical projection budget, the method achieves 71.2% of the performance gain of full stepwise correction using only 25% of the correction steps, substantially improving the trade-off between computational cost and sampling accuracy.

In scientific computing, generative models must strictly satisfy hard physical constraints—such as partial differential equations (PDEs)—yet gradient-based optimization is often infeasible due to gradient sparsity or high computational cost. Method: This paper proposes the Extrapolation–Correction–Interpolation (ECI) sampling framework, the first method to inject exact hard constraints into flow matching models *without gradients*, *without fine-tuning*, and *zero-shot*. Leveraging a pre-trained flow matching model, ECI iteratively refines generation trajectories via constraint-driven extrapolation, projection-based correction, and path interpolation—enforcing diverse PDE constraints rigorously while avoiding gradient computation and model adaptation. Contribution/Results: Experiments demonstrate that ECI achieves superior zero-shot generation quality over existing constrained generation baselines. Moreover, on regression tasks, it attains competitive accuracy without fine-tuning. ECI establishes an efficient, general-purpose paradigm for physics-informed generative modeling.

Existing time-series generation methods struggle to simultaneously satisfy hard constraints (e.g., power peak limits), ensure sample fidelity, and scale efficiently—particularly in engineering and safety-critical applications. To address this, we propose the Diffusion Posterior Projection Sampling (DPPS) framework: a plug-and-play, training-free approach that couples the posterior mean estimate of a pre-trained diffusion model with orthogonal projection onto the constraint set, enabling an iterative denoising–projection update scheme. We theoretically establish its convergence and, for the first time, achieve efficient, high-fidelity generation under hundreds of hard constraints. Evaluated on stock, traffic, and air quality datasets, DPPS improves sample quality by ~10% and temporal similarity by ~42% over state-of-the-art methods, while significantly enhancing robustness under stress testing and utility of privacy-preserving synthetic data.

This work addresses the challenge of enforcing complex nonlinear constraints—such as road-legal regions in robotic control and autonomous driving—within generative models, where existing approaches often fail to simultaneously ensure constraint satisfaction and high-fidelity generation. The authors propose a constrained fine-tuning framework that leverages pre-trained generative models to produce outputs strictly confined within structured feasible regions, without compromising sample realism. By overcoming the limitations of conventional fine-tuning or training-free strategies, the method achieves superior performance across diverse and intricate constraint scenarios, consistently outperforming current baselines in both generation quality and adherence to constraints.

Existing diffusion models often suffer from low constraint satisfaction rates or degraded sample quality when generating samples under complex feasibility constraints, primarily due to distributional mismatches between training and sampling phases. This work proposes a trajectory-aware fine-tuning framework based on online rollouts, which integrates constraint guidance during training and, for the first time, incorporates a numerical integration perspective into the diffusion process. By end-to-end differentiating denoising trajectories under a fixed noise schedule, the method explicitly exposes constraint violations, thereby aligning the training and sampling distributions. Combining constraint-aware guidance, differentiable noise scheduling, and an online rollout mechanism, the approach significantly improves constraint satisfaction across multiple tasks while maintaining generation quality on par with current state-of-the-art methods.

This study addresses the issue that enforcing constraints during sampling often causes severe deviations from the pretrained distribution. To this end, it proposes a training-free framework that formulates constraint enforcement as minimal intervention on flow trajectories. By deriving closed-form solutions via adjoint equations, the method eliminates iterative optimization overhead and adaptively selects optimal intervention timings to balance constraint satisfaction with distribution fidelity in perturbed intermediate states. This flow matching-based minimal trajectory intervention technique requires no additional training. It significantly improves constraint satisfaction rates across visual generation and physical system tasks while preserving generative distribution quality superior to existing state-of-the-art methods.

This work addresses the challenge of jointly achieving scenario optimization and conformal prediction with rigorous safety guarantees under limited sample sizes, while appropriately allocating risk across multi-output or multi-stage tasks. From a systems and control perspective, we introduce—for the first time—a natural integration of a sample removal mechanism into the conformal prediction framework, treating discarded samples as acceptable exceptions. We propose a modular risk allocation rule that composes multiple local calibration certificates to construct a unified joint guarantee. The approach leverages exchangeability to derive an average violation law and incorporates multi-step tube-based calibration, making it suitable for multi-output prediction and finite-horizon control. Numerical experiments demonstrate that the proposed strategy effectively balances performance and safety in constraint tightening problems.

本文提出了一种近似均匀采样器，用于解决满足特定条件的约束满足问题，通过调用一种高效的近似计数算法，在多项式时间内生成接近均匀分布的解。