In document-level relation extraction (RE), the models are required to correctly predict implicit relations in documents via relational reasoning. To this end, many graph-based methods have been proposed for this task. Despite their success, these methods still suffer from several drawbacks: 1) their interaction between document encoder and graph encoder is usually unidirectional and insufficient; 2) their graph encoders often fail to capture the global context of nodes in document graph. In this paper, we propose a document-level RE model with a Graph-Transformer Network (GTN). The GTN includes two core sublayers: 1) the graph-attention sublayer that simultaneously models global and local contexts of nodes in the document graph; 2) the cross-attention sublayer, enabling GTN to capture the non-entity clue information from the document encoder. Furthermore, we introduce two auxiliary training tasks to enhance the bidirectional semantic interaction between the document encoder and GTN: 1) the graph node reconstruction that can effectively train our cross-attention sublayer to enhance the semantic transition from the document encoder to GTN; 2) the structure-aware adversarial knowledge distillation, by which we can effectively transfer the structural information of GTN to the document encoder. Experimental results on four benchmark datasets prove the effectiveness of our model. Our source code is available at https://github.com/DeepLearnXMU/DocRE-BSI.