Precise and accurate predictions over boundary areas are essential for semantic segmentation. However, the commonly used convolutional operators tend to smooth and blur local detail cues, making it difficult for deep models to generate accurate boundary predictions. In this paper, we introduce an operator-level approach to enhance semantic boundary awareness, so as to improve the prediction of the deep semantic segmentation model. Specifically, we formulate the boundary feature enhancement process as an anisotropic diffusion process. We propose a novel learnable approach called semantic diffusion network (SDN) for approximating the diffusion process, which contains a parameterized semantic difference convolution operator followed by a feature fusion module and constructs a differentiable mapping from original backbone features to advanced boundary-aware features. The proposed SDN is an efficient and flexible module that can be plugged into existing encoder-decoder segmentation models. Extensive experiments show that our approach can achieve consistent improvements over several typical state-of-the-art segmentation baseline models on challenging public benchmarks.