We propose a general framework for conditional sampling in PDE-based inverse problems, targeting the recovery of whole solutions from extremely sparse or noisy measurements. This is accomplished by a function-space diffusion model and plug-and-play guidance for conditioning. Our method first trains an unconditional discretization-agnostic denoising model using neural operator architectures. At inference, we refine the samples to satisfy sparse observation data via a gradient-based guidance mechanism. Through rigorous mathematical analysis, we extend Tweedie's formula to infinite-dimensional Hilbert spaces, providing the theoretical foundation for our posterior sampling approach. Our method (FunDPS) accurately captures posterior distributions in function spaces under minimal supervision and severe data scarcity. Across five PDE tasks with only 3% observation, our method achieves an average 32% accuracy improvement over state-of-the-art fixed-resolution diffusion baselines while reducing sampling steps by 4x. Furthermore, multi-resolution fine-tuning ensures strong cross-resolution generalizability. To the best of our knowledge, this is the first diffusion-based framework to operate independently of discretization, offering a practical and flexible solution for forward and inverse problems in the context of PDEs. Code is available at https://github.com/neuraloperator/FunDPS

翻译：我们提出了一个基于偏微分方程（PDE）逆问题中条件采样的通用框架，旨在从极其稀疏或含噪声的测量中恢复完整解。该框架通过函数空间扩散模型和即插即用的条件引导机制实现。我们的方法首先使用神经算子架构训练一个与离散化无关的无条件去噪模型。在推理阶段，我们通过基于梯度的引导机制对样本进行优化，使其满足稀疏观测数据。通过严格的数学分析，我们将Tweedie公式推广到无限维希尔伯特空间，为我们的后验采样方法提供了理论基础。我们的方法（FunDPS）能够在极少量监督和严重数据稀缺的情况下，准确捕捉函数空间中的后验分布。在仅使用3%观测数据的五个PDE任务中，我们的方法相比最先进的固定分辨率扩散基线实现了平均32%的精度提升，同时将采样步骤减少了4倍。此外，多分辨率微调确保了强大的跨分辨率泛化能力。据我们所知，这是首个独立于离散化操作的基于扩散的框架，为PDE背景下的正问题和逆问题提供了实用且灵活的解决方案。代码可在https://github.com/neuraloperator/FunDPS获取。