DiffusionPDE is a generative model for solving partial differential equations (PDEs) under partial observation. The method addresses the challenge of solving PDEs when only partial information is available, by learning the joint distribution of the coefficient and solution spaces. It uses a diffusion model to generate plausible data, guided by sparse observations and PDE constraints. The model can simultaneously fill in missing information and solve PDEs, outperforming existing state-of-the-art methods in both forward and inverse directions. DiffusionPDE is tested on various PDEs, including Darcy Flow, Poisson, Helmholtz, Burger's, and Navier-Stokes equations, and shows significant improvements in accuracy and performance. The model is trained on a family of PDEs and can handle arbitrary sparsity patterns with a single pre-trained diffusion model. It is capable of recovering both the coefficient and solution spaces from sparse observations, and has been shown to effectively reconstruct the complete state of Burger's equation using time-series data from just five sensors. The method is also robust to different sampling patterns of sparse observations and can handle both static and dynamic PDEs. DiffusionPDE is a versatile framework for solving PDEs under partial observation, and has the potential to revolutionize physical modeling in real-world applications.DiffusionPDE is a generative model for solving partial differential equations (PDEs) under partial observation. The method addresses the challenge of solving PDEs when only partial information is available, by learning the joint distribution of the coefficient and solution spaces. It uses a diffusion model to generate plausible data, guided by sparse observations and PDE constraints. The model can simultaneously fill in missing information and solve PDEs, outperforming existing state-of-the-art methods in both forward and inverse directions. DiffusionPDE is tested on various PDEs, including Darcy Flow, Poisson, Helmholtz, Burger's, and Navier-Stokes equations, and shows significant improvements in accuracy and performance. The model is trained on a family of PDEs and can handle arbitrary sparsity patterns with a single pre-trained diffusion model. It is capable of recovering both the coefficient and solution spaces from sparse observations, and has been shown to effectively reconstruct the complete state of Burger's equation using time-series data from just five sensors. The method is also robust to different sampling patterns of sparse observations and can handle both static and dynamic PDEs. DiffusionPDE is a versatile framework for solving PDEs under partial observation, and has the potential to revolutionize physical modeling in real-world applications.