Deep generative models hold great promise for representing complex physical systems, but their deployment is currently limited by the lack of guarantees on the physical plausibility of the generated outputs. Ensuring that known physical constraints are enforced is therefore critical when applying generative models to scientific and engineering problems. We address this limitation by developing a principled framework for sampling from a target distribution while rigorously satisfying mathematical constraints. Leveraging the variational formulation of Langevin dynamics and Lagrangian duality, we propose Constrained Alternated Split Augmented Langevin (CASAL), a novel primal-dual sampling algorithm that enforces constraints progressively through variable splitting. We analyze our algorithm in Wasserstein space and derive explicit mixing time rates. While the method is developed theoretically for Langevin dynamics, we demonstrate its applicability to diffusion models. We apply our method to diffusion-based data assimilation on a complex physical system, where enforcing physical constraints substantially improves both forecast accuracy and the preservation of critical conserved quantities. We also demonstrate the potential of CASAL for challenging non-convex feasibility problems in optimal control.

翻译：深度生成模型在表示复杂物理系统方面展现出巨大潜力，但其应用目前受限于生成输出的物理合理性缺乏保证。因此，在将生成模型应用于科学与工程问题时，确保已知物理约束得到强制执行至关重要。我们通过开发一个从目标分布中采样同时严格满足数学约束的原则性框架来解决这一局限性。利用朗之万动力学的变分公式与拉格朗日对偶性，我们提出了约束交替分裂增广朗之万算法，这是一种新颖的原对偶采样算法，通过变量分裂逐步强制执行约束。我们在Wasserstein空间中分析该算法，并推导出显式的混合时间速率。虽然该方法在理论上针对朗之万动力学发展，但我们证明了其对于扩散模型的适用性。我们将该方法应用于复杂物理系统中基于扩散的数据同化问题，其中物理约束的强制执行显著提高了预报精度与关键守恒量的保持能力。我们还展示了约束交替分裂增广朗之万算法在最优控制中具有挑战性的非凸可行性问题上的潜力。