Molecular structure analysis is fundamental in biology, chemistry, and medicine. NMR is a promising technique for non-destructive structure determination under living conditions, but conventional NMR relies on large molecule ensembles, losing single-molecule information. Nanoscale magnetic resonance spectroscopy using nitrogen-vacancy (NV) centers offers single-spin sensitivity and single-molecule resolution. While nanoscale NMR allows insights into molecular structure, 2D nanoscale NMR is crucial but suffers from quadratically increasing measurement times with sampling numbers. Compressive sensing and sparse approximation have been applied, but compressive sensing requires prior knowledge, and sparse approximation has limited speed-up at low sampling coverage. Deep learning (DL), with its pattern recognition and noise cancellation capabilities, offers a potential solution. However, training DL networks on real experimental data is time-consuming. This paper proposes a DLMC method combining a deep learning network (trained on simulated data) and a matrix completion algorithm to reconstruct 2D NV spectra from sparsely sampled data, addressing the domain shift problem and capturing the global low-rank feature of the NV spectrum map.

Existing methods for accelerating 2D NMR include compressive sensing and sparse approximation. However, compressive sensing requires additional information for effective data recovery, limiting its applicability in certain scenarios. Sparse approximation techniques, while useful for accelerating 2D NMR, may struggle with very low sampling coverage and offer limited speed-up capabilities. Deep learning has shown promise in various fields, including super-resolution imaging, denoising, and inpainting, suggesting its potential for improving NMR spectroscopy. Recent work has explored the use of deep learning in physics, specifically for noise mitigation in nano-NMR and characterizing quantum dot arrays. However, the reliance on extensive experimental data for training DL models poses a significant challenge. Matrix completion techniques have also been used for handling missing data in various applications. Combining deep learning with matrix completion is a promising area with potential synergies.

The proposed DLMC method combines a deep learning convolutional neural network (DLNet) and a singular value thresholding (SVT) matrix completion (MC) algorithm. The DLNet is an encoder-decoder CNN designed for 2D NMR map reconstruction. The encoder extracts image features using convolutional layers and learnable pooling layers, while the decoder reconstructs the full-resolution map using transposed convolutional layers and residual blocks. Skipping connections between encoder and decoder layers preserve detailed information and mitigate gradient vanishing. The network is trained on simulated 2D NMR spectra, addressing the issue of limited experimental data. The training process involves padding the simulated map, normalization, noise addition, and L1 loss for backpropagation. The Adam optimizer with a learning rate of 5 x 10⁻⁴ is used for 500 epochs. The output of the DLNet is then post-processed using the SVT algorithm, a classical matrix completion method that enhances the low-rank property of the reconstructed map, further improving domain adaptation and alleviating the potential bias from the simulated training data. The DLMC algorithm consists of two steps: DLNet reconstruction followed by SVT post-processing. The nanoscale 2D NMR experiments are performed using a coupled nuclear cluster probed by an NV quantum sensor, utilizing a COSY-like protocol. Data are acquired by sweeping t₁ and t₂ parameters, with each entry averaged over 1.5 x 10⁵ measurements. The experimental time domain data is normalized to [0,1] before being processed by the DLMC algorithm. Different sampling coverages (10%, 40%, 80%) are used to evaluate the method. The simulation for training the DLNet is based on the Schrödinger equation with the NV-nuclear spin system Hamiltonian. The simulation considers the hyperfine coupling between the NV center and the nuclei and the dipolar coupling between the nuclei. The training data consists of pairs of partially sampled and full resolution maps. The building blocks of the DLNet, including convolutional layers, pooling layers, ReLU activation layers, and transposed convolutional layers are detailed. Robustness analysis is performed using multiple randomly generated sampling matrices with a fixed sampling coverage of 10%. Finally, the DLMC methodology is tested on simulated data for multiple nuclear spins.

The DLMC method significantly improves the reconstruction of 2D nanoscale NMR spectra from sparsely sampled data. For 10% sampling coverage, the DLMC method achieves an SNR enhancement of 5.7 ± 1.3 dB compared to the original data, significantly outperforming the MC-only method (3.2 ± 3.1 dB enhancement). Even at 10% sampling coverage, DLMC successfully recovers the spectrum, while the MC-only method fails. The RMSE analysis confirms that DLMC demonstrates the best overall performance. The method's robustness is demonstrated through testing with multiple randomly generated sampling matrices. Analysis shows that DLMC has smaller variances in RMSE and SNR, indicating better robustness compared to the MC-only method. The structural similarity index measure (SSIM) is introduced to evaluate the quality of the reconstruction, considering structural similarity between the original and reconstructed spectra. The DLMC method consistently shows improved SSIM values compared to the original under-sampled data. Finally, the DLMC approach is successfully tested on simulated data for a system with multiple (5) nuclear spins, demonstrating its general applicability.

The results demonstrate the effectiveness of the DLMC method in accelerating 2D nanoscale NMR spectroscopy. The combination of deep learning and matrix completion successfully addresses the challenges of limited experimental data and low sampling coverage. The significant improvement in SNR and RMSE, along with the robustness analysis, underscores the method's reliability. The inclusion of SSIM as an evaluation metric provides a more comprehensive assessment of the reconstruction quality, beyond simply noise reduction. The successful application of DLMC to a multi-nuclear spin system suggests broad applicability beyond the specific experimental setup. This advancement allows for a substantial reduction in experimental time, potentially by an order of magnitude, paving the way for high-throughput nanoscale NMR studies. The enhanced speed and sensitivity offer great potential for structural analysis at the single-molecule level, providing valuable insights into molecular interactions and dynamics.

This study demonstrates AI-enhanced nanoscale 2D NMR spectroscopy using an NV quantum sensor. Deep learning allows for the reconstruction of full 2D spectra from just 10% of the data, drastically reducing experimental time. The DLMC method significantly enhances the signal-to-noise ratio and reduces reconstruction error compared to other methods. Future work could explore incorporating SSIM into the loss function for further distortion reduction and investigating specialized sampling schemes for further optimization. The method shows potential for broader application in other magnetic resonance and imaging experiments, with the possibility of constructing 3D molecular structures from sufficient bond length and angle information.

While the DLMC method shows significant improvement, some artifacts and amplitude distortions remain, particularly at low sampling coverages. The training of the DLNet relies on simulated data, which may not perfectly represent the complexities of real experimental data. The study focuses on a specific experimental system, and further investigation is needed to determine its generalizability to other systems and molecules. The current implementation assumes a specific sampling strategy, and the performance might vary with different sampling patterns.