Sub-sampling of NMR Correlation and Exchange Experiments

Sub-sampling is applied to simulated $T_1$-$D$ NMR signals and its influence on inversion performance is evaluated. For this different levels of sub-sampling were employed ranging from the fully sampled signal down to only less than two percent of the original data points. This was combined with multiple sample schemes including fully random sampling, truncation and a combination of both. To compare the performance of different inversion algorithms, the so-generated sub-sampled signals were inverted using Tikhonov regularization, modified total generalized variation (MTGV) regularization, deep learning and a combination of deep learning and Tikhonov regularization. Further, the influence of the chosen cost function on the relative inversion performance was investigated. Overall, it could be shown that for a vast majority of instances, deep learning clearly outperforms regularization based inversion methods, if the signal is fully or close to fully sampled. However, in the case of significantly sub-sampled signals regularization yields better inversion performance than its deep learning counterpart with MTGV clearly prevailing over Tikhonov. Additionally, fully random sampling could be identified as the best overall sampling scheme independent of the inversion method. Finally, it could also be shown that the choice of cost function does vastly influence the relative rankings of the tested inversion algorithms highlighting the importance of choosing the cost function accordingly to experimental intentions.

Deep Learning as a Method for Inversion of NMR Signals

The concept of deep learning is employed for the inversion of NMR signals and it is shown that NMR signal inversion can be considered as an image-to-image regression problem, which can be treated with a convolutional neural net. It is further outlined, that inversion through deep learning provides a clear efficiency and usability advantage compared to regularization techniques such as Tikhonov and modified total generalized variation (MTGV), because no hyperparemeter selection prior to reconstruction is necessary. The inversion network is applied to simulated NMR signals and the results compared with Tikhonov- and MTGV-regularization. The comparison shows that inversion via deep learning is significantly faster than the latter regularization methods and also outperforms both regularization techniques in nearly all instances.

Bayesian Inference for NMR Spectroscopy with Applications to Chemical Quantification

Nuclear magnetic resonance (NMR) spectroscopy exploits the magnetic properties of atomic nuclei to discover the structure, reaction state and chemical environment of molecules. We propose a probabilistic generative model and inference procedures for NMR spectroscopy. Specifically, we use a weighted sum of trigonometric functions undergoing exponential decay to model free induction decay (FID) signals. We discuss the challenges in estimating the components of this general model -- amplitudes, phase shifts, frequencies, decay rates, and noise variances -- and offer practical solutions. We compare with conventional Fourier transform spectroscopy for estimating the relative concentrations of chemicals in a mixture, using synthetic and experimentally acquired FID signals. We find the proposed model is particularly robust to low signal to noise ratios (SNR), and overlapping peaks in the Fourier transform of the FID, enabling accurate predictions (e.g., 1% sensitivity at low SNR) which are not possible with conventional spectroscopy (5% sensitivity).