Normally, these models work by inverting the process of natural diffusion, where they start with a distribution of random noise and progressively transform it into a structured data distribution resembling the training data. This transformation occurs in multiple steps, which incrementally denoise the noisy sample until it reaches the desired complexity and detail. In contrast to the normal diffusion models mentioned above (Song et al. 2022, 2024), which generate synthetic images by denoising random noise distributions without incorporating any specific guidance, our GenMDI model generates a synthetic image considering the previous image and the next image surrounding the generated image. This image generation process with guidance or condition is known as the conditional diffusion process, which is often used in the generation of video frames (Voleti et al. 2022). By conditioning the reverse diffusion process on the previous and subsequent images, GenMDI ensures that the generated image maintains continuity and reflects the dynamics of the surrounding images. This approach not only preserves the natural flow and consistency of MDI time-series magnetograms but also enhances our model's ability to accurately generate synthetic images. To our knowledge, this is the first time a conditional diffusion model has been used to improve the temporal resolution of MDI magnetograms. The remainder of this paper is organized as follows. Section 2 describes the data used in this study.

Deep learning, which is a subfield of machine learning, has drawn significant interest in recent years. It was originally used in speech recognition (Deng, Hinton, and Kingsbury, 2013), natural language processing (Kastrati et al., 2021), and computer vision (Hu et al., 2018). More recently, it has been applied to astronomy, astrophysics, and solar physics (Liu et al., 2020a; Jiang et al., 2021; Espuña Fontcuberta et al., 2023; Mercea et al., 2023; Scully et al., 2023). Here, we present a new deep-learning method, specifically an attention-aided convolutional neural network (CNN), named SolarCNN, for solar image super-resolution. SolarCNN aims to enhance the quality of line-of-sight (LOS) magnetograms of solar active regions (ARs) collected by the Michelson Doppler Imager (MDI; Scherrer et al., 1995) on board the Solar and Heliospheric Observatory (SOHO; Domingo, Fleck, and Poland, 1995). The ground-truth labels used for training SolarCNN are the LOS magnetograms of the same ARs collected by the Helioseismic and Magnetic Imager (HMI; Schou et al., 2012) on board the Solar Dynamics Observatory (SDO; Pesnell, Thompson, and Chamberlin, 2012). Training and test samples are collected from ARs in the HMI and MDI overlap period, between 1 May 2010 and 11 April 2011. An AR on the solar disk usually consists of one or more sunspots and pores that are formed because of the concentrations of strong magnetic fields.