发表机构
Umeå University; Università Campus Bio-Medico di Roma（于默奥大学; 罗马生物医学自由大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对流匹配模型生成物理状态时偏离目标分布的问题，提出源锚定方法，在生成前将物理约束编码进源噪声，在多个PDE系统上最准确重现目标分布且匹配约束精度。
AI 中文摘要
深度生成模型被用于求解偏微分方程并对物理系统状态的分布进行建模，但确保生成的样本满足支配定律仍然具有挑战性。基于投影的流匹配方法通过从未受约束的噪声分布中校正流来强制执行物理约束。这些校正会将生成的样本移离目标解的分布，尤其是在高噪声区域。为解决这一局限，我们提出了用于物理一致性的源锚定（SAPC），这是一种函数流匹配方法，在生成开始之前将物理约束编码到源噪声中。我们在五个由偏微分方程支配的系统上评估了SAPC，涵盖了具有线性和非线性动力学的六项任务，并将结果与五个基线以及无约束骨干模型进行了比较。锚定源减少了将样本驱动到可接受但偏离分布状态所需的大规模校正，并且SAPC在每项评估任务上都能最准确地重现目标分布，同时匹配最佳基于投影基线的约束精度。消融实验表明，这一增益源于将源投影与匹配的训练目标相结合，该目标向投影后的源进行回归。这些结果确定了源分布是物理一致生成建模的关键设计选择。
英文摘要
Deep generative models are used to solve partial differential equations and model distributions of physical system states, but ensuring that the generated samples satisfy the governing laws remains challenging. Projection-based flow-matching methods enforce physics by correcting the flow from an unconstrained noise distribution. These corrections shift the generated samples away from the distribution of target solutions, especially in high noise regions. To address this limitation, we propose Source Anchoring for Physical Consistency (SAPC), a Functional Flow Matching method that encodes the physical constraints into the source noise before generation begins. We evaluate SAPC on five systems governed by partial differential equations, covering six tasks with linear and non-linear dynamics, and compare results against five baselines and the unconstrained backbone. Anchoring the source reduces the need for large corrections that drive samples onto admissible but off-distribution states, and SAPC reproduces the target distributions most accurately on every evaluated task, while matching the constraint precision of the best projection-based baselines. Ablation experiments show that this gain arises from pairing source projection with a matched training objective that regresses toward the projected source. These results identify the source distribution as a key design choice for physically consistent generative modelling.
CommentsSubmitted to ICLR 2027
