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Nuclear magnetic resonance (NMR) spectroscopy serves as an indispensable tool in chemistry and biology but often suffers from long experimental time. We present a proof-of-concept of application of deep learning and neural network for highquality, reliable, and very fast NMR spectra reconstruction from limited experimental data. We show that the neural network training can be achieved using solely synthetic NMR signal, which lifts the prohibiting demand for large volume of realistic training data usually required in the deep learning approach.Nuclear magnetic resonance (NMR) spectroscopy is an invaluable biophysical tool in modern chemistry and life sciences. Examples include characterization of complex protein structures 1, 2 and studies disordered 3 and short-lived molecular systems 4 . However, duration of NMR experiments increase rapidly with spectral resolution and dimensionality 5, 6 , which often imposes unbearable limitations due to low sample stability and/or excessive costs of NMR measurement time. To accelerate the data acquisition and optimize sensitivity, modern NMR experiments are often acquired using the Non-Uniform Sampling (NUS) approach, where only a small fraction of traditional NMR measurements, usually called free induction decay (FID), is performed and, thus, only a fraction of measurement time is spent.Over the past two decades, several methods 5, 7, 8, 9, 10, 11, 12 have been established in the NMR field to reconstruct high quality spectra from NUS data. In all case, a prior knowledge or assumption are incorporated in order to compensate for missing information introduced by the NUS scheme. Examples include the maximum entropy 6 , spectrum sparsity in compressed sensing 9, 10, 13, 14 , spectral line-shape estimation in SMILE 15 , tensor structures in MDD 5 or Hankel tensors 11 , and exponential nature of NMR signal in low rank 7 . Thus, although spectra are reconstructed well with these approaches, a number of important practical limitations and conceptual question remain. Thus, despite of varying implementations, algorithms of all these methods are iterative and require lengthy calculations and/or use of super-computers. Pros and cons of applying different prior assumptions are not well understood and combination of the best features, while avoiding the negative sides of different approaches is problematic.Motivated by the exciting achievements of deep learning (DL) 16, 17 , a representative artificial intelligence using neural networks, we will explore the end-to-end mapping with DL for the NMR spectra reconstruction, enabling fast and high-quality reconstructions. In contrast to the traditional methods that take advantage of one or more predefined priors for reconstruction, for instance, sparsity and low rank, the proposed DL approach mines the underlying information embedded in data and thus does not require any predefined priors.A critical challenge of the DL is that it requires an enormous amount of realistic experimental data at the training stage.Whilst obtaining of such a gigan...

We present a novel analysis of the boundary integral operators associated to the wave equation. The analysis is done entirely in the time-domain by employing tools from abstract evolution equations in Hilbert spaces and semi-group theory. We prove a single general theorem from which well-posedness and regularity of the solutions for several boundary integral formulations can be deduced as particular cases. By careful choices of continuous and discrete spaces, we are able to provide a concise analysis for various direct and indirect formulations, both at the continuous level and for their Galerkin-in-space semi-discretizations. Some of the results here are improvements on previously known results, while other results are equivalent to those in the literature. The methodology presented here greatly simplifies the analysis of the operators of the Calderón projector for the wave equation and can be generalized for other relevant boundary integral equations.

Nuclear magnetic resonance (NMR) spectroscopy serves as an indispensable tool in chemistry and biology but often suffers from long experimental times. We present a proof‐of‐concept of the application of deep learning and neural networks for high‐quality, reliable, and very fast NMR spectra reconstruction from limited experimental data. We show that the neural network training can be achieved using solely synthetic NMR signals, which lifts the prohibiting demand for a large volume of realistic training data usually required for a deep learning approach.

Low-intensity signal reconstruction is generally challenging in biological MRS and we provide a solution to this problem. The proposed method may be extended to recover signals that generally can be modeled as a sum of exponential functions in biomedical engineering applications, e.g., signal enhancement, feature extraction, and fast sampling.