Excited to introduce our latest work, Guided Diffusion Sampling on Function Spaces (FunDPS) (arxiv.org/abs/2505.17004) - a discretization-agnostic generative framework for solving PDE-based forward and inverse problems. Diffusion-based posterior sampling on function spaces: Our model recovers full-field PDE solutions, coefficient functions, and boundary conditions from severely sparse (just 3%) measurements, yielding SotA performance in both speed and accuracy. Multi-resolution operator learning pipeline: FunDPS leverages Gaussian Random Field priors and neural operator architectures, enabling multi-resolution training and inference, reducing training time by 25% and inference time by 50%. Infinite-dimensional Tweedie’s Formula: We extend Tweedie’s formula into infinite-dimensional Banach spaces, forming the rigorous theoretical foundation for posterior mean estimation. Results: Achieved an average 32% accuracy improvement and 4x fewer sampling steps compared to previous SOTA approaches across five challenging PDE tasks. Plus, our multi-resolution inference pipeline accelerates computations by up to 25x! Paper (arxiv.org/abs/2505.17004). Code (github.com/neuraloperator/Fu…), based on our earlier workshop paper (ml4physicalsciences.github.i…). @jiacheny7, @AbbasMammadov11, @julberner, @gavinkerrigan, Jong Chul Ye, @Azizzadenesheli #DiffusionModels #InverseProblems #PDE #MachineLearning #NeuralOperators #AI4Science