The fusion of the low-frequency subband in the non-subsampled shearlet transform (NSST) domain requires artificially obtained fusion modes; thus, the spatial continuity and contour detail information of the source image are not adequately captured. An infrared and visible image fusion algorithm based on a convolutional neural network is proposed to solve this problem. First, the Siamese convolutional neural network is used to learn the characteristics of the low-frequency subband in the NSST domain and output a feature map that measures the spatial detail information of the subbands. Then, on the basis of the feature map obtained by Gaussian filter processing, a local-similarity-based measurement function is designed to adaptively adjust the fusion mode of the low-frequency subband in the NSST domain. Finally, on the basis of the variance of the high-frequency subband in the NSST domain, the local region energy, and the visibility characteristics, the pulse-coupled neural network (PCNN) parameters are adaptively set to complete the fusion of the high-frequency subband in the NSST domain. Experimental results show that the QAB/F index of the algorithm is slightly lower than that of the comparison algorithm. However, the spatial frequency, SP, structural similarity, and visual information fidelity for fusion are improved by approximately 50.42%, 14.25%, 7.91%, and 61.67%, respectively, which indicates that the method effectively solves the low-frequency subband fusion mode. It also eliminates the need to manually set the PCNN parameters to solve this problem.