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

📄 arXiv: 2505.22391v1 📥 PDF

作者: Yi Zhang, Difan Zou

分类: cs.LG, cs.AI, cs.CE, math.NA

发布日期: 2025-05-28

备注: 23 pages, 5 figures, 4 tables

💡 一句话要点

提出物理信息蒸馏方法以解决扩散模型中的PDE约束问题

🎯 匹配领域: 支柱二：RL算法与架构 (RL & Architecture)

关键词: 扩散模型 物理信息蒸馏 偏微分方程 生成建模 正逆问题 计算机视觉 机器学习

📋 核心要点

  1. 现有扩散模型在处理PDE约束时面临挑战，无法直接在干净样本上施加约束，导致生成精度下降。
  2. 本文提出物理信息蒸馏扩散模型（PIDDM），通过后处理蒸馏阶段施加PDE约束，避免了直接在扩散过程中注入约束的复杂性。
  3. 实验结果显示，PIDDM在多个PDE基准测试中显著提高了PDE约束的满足度，相较于PIDM、DiffusionPDE等基线方法，计算开销更低。

📝 摘要（中文）

以生成方式建模物理系统具有处理部分观测、生成多样解和解决正逆问题的优势。近年来，扩散模型在物理系统建模中受到关注，但由于只能访问噪声数据，直接施加约束在干净样本上存在困难。为此，本文提出了一种后处理蒸馏方法——物理信息蒸馏扩散模型（PIDDM），在蒸馏阶段施加PDE约束，显著提高了PDE的满足度，并支持正逆问题的求解。实验结果表明，PIDDM在多个PDE基准测试中优于现有方法，且计算开销更小。

🔬 方法详解

问题定义：本文旨在解决扩散模型在处理物理系统时，因只能访问噪声数据而无法直接施加PDE约束的问题。现有方法通常通过期望值施加约束，但这会导致Jensen间隙，影响生成精度。

核心思路：提出物理信息蒸馏扩散模型（PIDDM），在后处理阶段施加PDE约束，而非在扩散过程中直接注入，从而提高生成样本的PDE满足度。

技术框架：PIDDM的整体架构包括两个主要阶段：扩散生成阶段和后处理蒸馏阶段。在扩散生成阶段，模型生成噪声样本；在后处理蒸馏阶段，施加PDE约束以优化生成结果。

关键创新：PIDDM的创新在于将PDE约束的施加从扩散过程转移到后处理阶段，避免了直接施加约束带来的复杂性，同时提高了生成样本的质量。

关键设计：在设计中，PIDDM使用了学习的评分网络来估计样本的期望值，并在蒸馏阶段通过特定的损失函数来优化PDE约束的满足度，确保生成样本的物理一致性。

🖼️ 关键图片

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📊 实验亮点

实验结果表明，PIDDM在多个PDE基准测试中显著提高了PDE约束的满足度，相比于PIDM、DiffusionPDE等基线方法，提升幅度达到XX%，且计算开销减少了YY%。

🎯 应用场景

该研究在物理系统建模、工程模拟和科学计算等领域具有广泛的应用潜力。通过有效地将物理约束融入生成模型，PIDDM能够在处理复杂物理现象时提供更准确的解决方案，推动相关领域的研究与应用发展。

📄 摘要（原文）

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.