Laser speckle is a long-standing issue which will corrupt image formation and interpretation. Over the years, various techniques have been developed to mitigate the speckle noise. Although machine learning approaches have been widely applied for image denoising tasks, there are very few of them are specifically designed for speckle reduction. In this work, we present a network specialized for reducing speckle noise in images. The training set consists of nearly 3,000 coherent- and incoherent-illuminated image pairs of a variety objects. The network is trained to learn a transformation from speckled to speckle-free images. We compare against the traditional image processing methods, and show that our learning-based approach outperforms these methods regarding both high PSNR, SSIM results and maintaining high-frequency edge features. Our data-driven approach dramatically reduces laser speckle noise by 11.71 dB, compared to a 0.17 dB reduction from non-local means filtering, a 0.1 dB reduction from median filtering and a 0.12 dB reduction from Gaussian filtering. Moreover, conventional image processing approaches reduce both laser speckle and high-frequency image features, which will result in the blurring effect. In contrast with optical speckle-denoising approaches, our method reduces cost and computational complexity. So, applications that require small and bright illumination sources with high-quality imaging can benefit from our work.