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Generating multi-contrasts/modal MRI of the same anatomy enriches diagnostic information but is limited in practice due to excessive data acquisition time. In this paper, we propose a novel deep-learning model for joint reconstruction and synthesis of multi-modal MRI using incomplete k-space data of several source modalities as inputs. The out- put of our model includes reconstructed images of the source modalities and high-quality image synthesized in the target modality. Our pro- posed model is formulated as a variational problem that leverages several learnable modality-specific feature extractors and a multimodal synthesis module. We propose a learnable optimization algorithm to solve this model, which induces a multi-phase network whose parameters can be trained using multi-modal MRI data. Moreover, a bilevel-optimization framework is employed for robust parameter training. We demonstrate the effectiveness of our approach using extensive numerical experiments. 

An efficient, durable, and low‐cost hydrogen evolution reaction (HER) catalyst is an essential requirement for practical hydrogen production. Herein, an effective approach to facilitate the HER kinetics of molybdenum carbide (Mo2C) electrocatalysts is presented by tuning its electronic structure through atomic engineering of nitrogen implantation. Starting from the organoimido‐derivatized polyoxometalate nanoclusters with inherent Mo-N bonds, the formation of N‐implanted Mo2C (N@Mo2C) nanocrystals with perfectly adjustable amounts of N atoms is demonstrated. The optimized N@Mo2C electrocatalyst exhibits remarkable HER performance and good stability over 20 h in both acid and basic electrolytes. Further density functional theory calculations show that engineering suitable nitrogen atoms into Mo2C can regulate its electronic structure well and decrease Mo-H strength, leading to a great enhancement.