Named entity recognition (NER) is a widely studied task in natural language processing. Recently, a growing number of studies have focused on the nested NER. The span-based methods, considering the entity recognition as a span classification task, can deal with nested entities naturally. But they suffer from the huge search space and the lack of interactions between entities. To address these issues, we propose a novel sequence-to-set neural network for nested NER. Instead of specifying candidate spans in advance, we provide a fixed set of learnable vectors to learn the patterns of the valuable spans. We utilize a non-autoregressive decoder to predict the final set of entities in one pass, in which we are able to capture dependencies between entities. Compared with the sequence-to-sequence method, our model is more suitable for such unordered recognition task as it is insensitive to the label order. In addition, we utilize the loss function based on bipartite matching to compute the overall training loss. Experimental results show that our proposed model achieves state-of-the-art on three nested NER corpora: ACE 2004, ACE 2005 and KBP 2017. The code is available at https://github.com/zqtan1024/sequence-to-set.