标题： 基于物理信息的扩散模型蒸馏用于偏微分方程约束生成 显示英文标题

标题： Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generation

摘要： 以生成方式建模物理系统具有多个优势，包括能够处理部分观测值、生成多样化解决方案以及解决正向和反向问题。 最近，扩散模型在物理系统的建模中引起了越来越多的关注，特别是那些受偏微分方程（PDEs）支配的系统。 然而，扩散模型仅在中间步骤访问噪声数据$\boldsymbol{x}_t$，这使得直接在每个噪声级别上的干净样本$\boldsymbol{x}_0$上施加约束变得不可行。 作为变通方法，通常会将约束应用于干净样本$\mathbb{E}[\boldsymbol{x}_0|\boldsymbol{x}_t]$的期望值，该期望值是使用学习到的得分网络估计的。 然而，在期望值上施加PDE约束并不能严格表示在真实干净数据上的约束，这被称为詹森差距。 这种差距造成了权衡：施加PDE约束可能会以降低生成建模准确性为代价。 为了解决这个问题，我们提出了一种简单但有效的后蒸馏方法，其中PDE约束不会直接注入扩散过程，而是在后蒸馏阶段强制执行。 我们将这种方法称为物理信息引导的扩散模型后蒸馏（PIDDM）。 这种蒸馏不仅促进了单步生成，提高了PDE满足度，还支持正向和反向问题求解以及从随机部分观测值中重建。 广泛的实验表明，在各种PDE基准测试中，PIDDM显著提高了PDE满足度，超过了几个近期和竞争性的基线，例如PIDM、DiffusionPDE和ECI采样，同时计算开销更小。 我们的方法可以为更高效和有效的方法提供指导，以将物理约束纳入扩散模型中。 显示英文摘要

摘要： Modeling physical systems in a generative manner offers several advantages, including the ability to handle partial observations, generate diverse solutions, and address both forward and inverse problems. Recently, diffusion models have gained increasing attention in the modeling of physical systems, particularly those governed by partial differential equations (PDEs). However, diffusion models only access noisy data $\boldsymbol{x}_t$ at intermediate steps, making it infeasible to directly enforce constraints on the clean sample $\boldsymbol{x}_0$ at each noisy level. As a workaround, constraints are typically applied to the expectation of clean samples $\mathbb{E}[\boldsymbol{x}_0|\boldsymbol{x}_t]$, which is estimated using the learned score network. However, imposing PDE constraints on the expectation does not strictly represent the one on the true clean data, known as Jensen's Gap. This gap creates a trade-off: enforcing PDE constraints may come at the cost of reduced accuracy in generative modeling. To address this, we propose a simple yet effective post-hoc distillation approach, where PDE constraints are not injected directly into the diffusion process, but instead enforced during a post-hoc distillation stage. We term our method as Physics-Informed Distillation of Diffusion Models (PIDDM). This distillation not only facilitates single-step generation with improved PDE satisfaction, but also support both forward and inverse problem solving and reconstruction from randomly partial observation. Extensive experiments across various PDE benchmarks demonstrate that PIDDM significantly improves PDE satisfaction over several recent and competitive baselines, such as PIDM, DiffusionPDE, and ECI-sampling, with less computation overhead. Our approach can shed light on more efficient and effective strategies for incorporating physical constraints into diffusion models.