Channel knowledge map (CKM) is a promising technique that enables environment-aware wireless networks by utilizing location-specific channel prior information to improve communication and sensing performance. A fundamental problem for CKM construction is how to utilize partially observed channel knowledge data to reconstruct a complete CKM for all possible locations of interest. This problem resembles the long-standing ill-posed inverse problem, which tries to infer from a set of limited observations the cause factors that produced them. By utilizing the recent advances of solving inverse problems with generative artificial intelligence (AI), in this paper, we propose generative CKM construction method using partially observed data by solving inverse problems with diffusion models. Simulation results show that the proposed method significantly improves the performance of CKM construction compared with benchmarking schemes.