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Phase recovery is a well-known inverse problem prevalent across science disciplines and attracts active research interests to develop a number of theoretical and experimental methods. Recent developments in artificial intelligence have further prompted research activities in processing the experimentally collected imperfect data, but applications have been limited to slow-varying data such as real images. Experimental noise present in largely fluctuating diffraction data, in particular, adds practical challenges to hamper consistent phase recovery. Here, we introduce a convolutional neural-network assisted $k$-space denoising method that can directly manage noisy diffraction signals. It showed superior performance on denoising the diffraction data, which promote improved phase recovery from noise-buried single-pulse diffraction signals obtained by the x-ray free-electron laser. Adapting our method to general diffraction data can expand boundaries of interpretable data and enhance observability of faint objects with weak signals.

Interest in atomic scale structures of individual specimens has invigorated developments of high-resolution probes, which include single-particle imaging using x-ray free-electron lasers (XFELs). The demonstrated spatial resolution, however, remains at tens of nanometers with difficulty in collecting diffraction signals at high frequency distinguished from noises. As such, various resolution-enhancement methods have been introduced, but few experimental verifications are available. Here, by carrying out XFEL single-pulse diffraction experiments, we explicitly unveil the dependence of SNRs on incident x-ray flux, data averaging, or multiparticle interference. We further propose a data-accumulation method of resolution-shell averaging as a robust scheme to improve the SNR. This study establishes a roadmap with which high-resolution XFEL single-pulse experiments can be contrived.