分离扩散采样提升逆问题求解效率与精度
Decoupled Diffusion Sampling for Inverse Problems on Function Spaces
- 解耦系数先验与物理过程，分别用无条件扩散和神经算子建模
- 稀疏观测下$ l_2 $误差降低11%，谱误差减少54%，数据仅1%时仍保持40%优势
- 适合物理信息少样本场景，尤其对逆向偏微分方程求解有显著帮助
我们提出一种高效、物理感知的函数空间生成框架，用于求解逆偏微分方程问题。现有插件式扩散后验采样器通过联合系数-解映射隐式表示物理规律，需大量成对监督数据。相比之下，我们的解耦扩散逆解器（DDIS）采用解耦设计：无条件扩散学习系数先验，神经算子显式建模前向偏微分方程以提供引导。该解耦结构实现更优的数据效率与物理信息学习能力，并自然支持解耦退火后验采样（DAPS），避免扩散后验采样中的过平滑问题。理论上，我们证明了当训练数据稀缺时，DDIS可避免联合模型的引导衰减失效。实验上，DDIS在稀疏观测下表现领先，平均$ l_2 $误差降低11%，谱误差降低54%；当数据仅占1%时，$ l_2 $误差仍比联合模型高40%。
原文摘要 · Abstract (English)
We propose a data-efficient, physics-aware generative framework in function space for inverse PDE problems. Existing plug-and-play diffusion posterior samplers represent physics implicitly through joint coefficient-solution modeling, requiring substantial paired supervision. In contrast, our Decoupled Diffusion Inverse Solver (DDIS) employs a decoupled design: an unconditional diffusion learns the coefficient prior, while a neural operator explicitly models the forward PDE for guidance. This decoupling enables superior data efficiency and effective physics-informed learning, while naturally supporting Decoupled Annealing Posterior Sampling (DAPS) to avoid over-smoothing in Diffusion Posterior Sampling (DPS). Theoretically, we prove that DDIS avoids the guidance attenuation failure of joint models when training data is scarce. Empirically, DDIS achieves state-of-the-art performance under sparse observation, improving $l_2$ error by 11% and spectral error by 54% on average; when data is limited to 1%, DDIS maintains accuracy with 40% advantage in $l_2$ error compared to joint models.
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