The paper introduces DiffusionPDE, a novel framework for solving partial differential equations (PDEs) using generative diffusion models. The approach addresses the challenge of incomplete or sparse observations, which are common in real-world applications. DiffusionPDE models the joint distribution of the solution and coefficient spaces, allowing it to fill in missing information and solve PDEs accurately. The method is evaluated on various static and dynamic PDEs, including Darcy Flow, Poisson, Helmholtz, Burger's, and Navier-Stokes equations. Extensive experiments demonstrate that DiffusionPDE outperforms state-of-the-art methods in both forward and inverse problems under sparse observations, achieving comparable results with full observations. The framework's versatility and effectiveness are highlighted through detailed results and comparisons with other PDE solvers, such as PINO, DeepONet, PINNs, and FNO.The paper introduces DiffusionPDE, a novel framework for solving partial differential equations (PDEs) using generative diffusion models. The approach addresses the challenge of incomplete or sparse observations, which are common in real-world applications. DiffusionPDE models the joint distribution of the solution and coefficient spaces, allowing it to fill in missing information and solve PDEs accurately. The method is evaluated on various static and dynamic PDEs, including Darcy Flow, Poisson, Helmholtz, Burger's, and Navier-Stokes equations. Extensive experiments demonstrate that DiffusionPDE outperforms state-of-the-art methods in both forward and inverse problems under sparse observations, achieving comparable results with full observations. The framework's versatility and effectiveness are highlighted through detailed results and comparisons with other PDE solvers, such as PINO, DeepONet, PINNs, and FNO.