Integrating the brain structural and functional connectivity features is of great significance in both exploring brain science and analyzing cognitive impairment clinically. However, it remains a challenge to effectively fuse structural and functional features in exploring the brain network. In this paper, a novel brain structure-function fusing-representation learning (BSFL) model is proposed to effectively learn fused representation from diffusion tensor imaging (DTI) and resting-state functional magnetic resonance imaging (fMRI) for mild cognitive impairment (MCI) analysis. Specifically, the decomposition-fusion framework is developed to first decompose the feature space into the union of the uniform and the unique spaces for each modality, and then adaptively fuse the decomposed features to learn MCI-related representation. Moreover, a knowledge-aware transformer module is designed to automatically capture local and global connectivity features throughout the brain. Also, a uniform-unique contrastive loss is further devised to make the decomposition more effective and enhance the complementarity of structural and functional features. The extensive experiments demonstrate that the proposed model achieves better performance than other competitive methods in predicting and analyzing MCI. More importantly, the proposed model could be a potential tool for reconstructing unified brain networks and predicting abnormal connections during the degenerative processes in MCI.