Zi Wang^{1}, Yihui Huang^{1}, Zhangren Tu^{2}, Di Guo^{2}, Vladislav Orekhov^{3}, and Xiaobo Qu^{1}

^{1}Department of Electronic Science, National Institute for Data Science in Health and Medicine, Xiamen University, Xiamen, China, ^{2}School of Computer and Information Engineering, Xiamen University of Technology, Xiamen, China, ^{3}Department of Chemistry and Molecular Biology, University of Gothenburg, Gothenburg, Sweden

Multi-dimensional nuclear magnetic resonance (NMR) spectroscopy is an invaluable biophysical tool but often suffers from long measurement. Several methods have been established for spectra reconstruction from undersampled data, two of which are model-based optimization and data-driven deep learning. Combining the main merits of them, we present a model-inspired flexible deep learning framework, for reliable, robust, and ultra-fast spectra reconstruction. Besides, we demonstrate that the model-inspired network needs very few parameters and is not sensitive to training datasets, which greatly reduces the demand for memory footprints and can work effectively in a wide range of scenarios without re-training.

Once the overall number of iterations is fixed, the data flow can be viewed as an unfolded deep learning network, as shown in Figure 1. Same to Eq. (1a), the spectrum is forced to maintain the data consistency to the sampled signal. Since the proper choice of thresholds is still of great demand and challenge, thus, instead of Eq. (1b), we use a learnable network $$$LS$$$ which can change thresholds with the characteristics of the input data, for adaptive soft-thresholding. The single network $$$LS$$$ is composed of convolutional layers, fully-connected layers, and a soft-thresholding. The overall number of iterations in our implementation is 10. With the increase of iterations, artifacts are gradually removed, and finally a high-quality reconstructed spectrum can be obtained.

Given a proof-of-concept of training neural networks using solely synthetic data with the exponential functions has been presented in paper

The reconstruction of 2D spectra in Figure 2 shows that, (a) MoDern can faithfully reconstruct it using 20% NUS data, and its peak intensity correlation reached 0.9998 with high fidelity of the lineshape reconstruction. (b) MoDern is comparable with, or may even surpass the state-of-the-art reconstruction method DLNMR

The most important advantage is that, MoDern abandons a large number of redundant convolution layers, which is often used in the data-driven deep learning

This work was supported in part by the National Natural Science Foundation of China (NSFC) under grants 61971361, 61871341, and U1632274, the Joint NSFC-Swedish Foundation for International Cooperation in Research and Higher Education (STINT) under grant 61811530021, the National Key R&D Program of China under grant 2017YFC0108703, the Natural Science Foundation of Fujian Province of China under grant 2018J06018, the Fundamental Research Funds for the Central Universities under grant 20720180056, the Xiamen University Nanqiang Outstanding Talents Program, the Science and Technology Program of Xiamen under grant 3502Z20183053, the Swedish Research Council under grant 2015–04614, and the Swedish Foundation for Strategic Research under grant ITM17-0218.

The correspondence should be sent to Prof. Xiaobo Qu (Email: quxiaobo@xmu.edu.cn)

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