Phase retrieval is crucial in various imaging techniques, including BCDI, where the phase of the scattered X-ray wave is lost during measurement. Recovering this phase information is essential for reconstructing 3D images and obtaining strain maps within the sample. Traditional iterative phase retrieval methods are computationally expensive, limiting real-time feedback in in-situ experiments. While deep learning offers faster alternatives, existing models typically require large datasets of labeled data, which are difficult and computationally expensive to obtain. This paper addresses these challenges by presenting AutoPhaseNN, an unsupervised physics-aware deep convolutional neural network. By integrating the X-ray scattering physics into the network design and training process, AutoPhaseNN learns to directly map diffraction intensities to real-space images without requiring paired input-output training data. This unsupervised approach eliminates the need for computationally expensive pre-processing of experimental data or the generation of simulated data sets, thus greatly enhancing efficiency and applicability. The paper highlights the model's application to 3D BCDI, emphasizing its potential for handling the large data volumes generated by advanced light sources, as well as its broader applicability to other phase retrieval problems where the forward model is known.

Existing deep learning approaches for phase retrieval are largely supervised, requiring large quantities of labeled data pairing diffraction patterns with corresponding real-space images. Obtaining this training data is either impractical (requiring computationally intensive iterative phase retrieval on experimental datasets) or leads to models that underperform on real-world data due to the discrepancies between simulated and experimental data. While generative adversarial networks (GANs) have been explored for unsupervised learning, they still rely on ground truth data for training. Physics-informed neural networks (PINNs) offer an alternative unsupervised approach, but have shown reduced accuracy compared to supervised methods in phase retrieval tasks. AutoPhaseNN addresses these limitations by combining a deep learning framework with explicit incorporation of the X-ray scattering physics to facilitate unsupervised training.

AutoPhaseNN combines a 3D convolutional encoder-decoder neural network with a physical model of X-ray scattering. The network architecture uses convolutional autoencoders and deconvolutional decoders with various layers (convolution, max pooling, upsampling, and zero padding) to learn the mapping between diffraction intensities and real-space amplitude and phase. A key aspect is the integration of the X-ray scattering forward model, which allows the network to generate estimated diffraction patterns based on its prediction. The model is trained using a loss function that minimizes the mean absolute error (MAE) between the measured diffraction intensities and those estimated by the network. Crucially, only the measured diffraction intensities are used for training; no real-space images are provided as ground truth during the training phase. The training data consists of both simulated data generated from atomistic simulations of gold crystals and a small amount of experimentally acquired BCDI data. The simulated data provides a broad range of samples for the network training, and the experimental data adds a crucial component to the model that accounts for imperfections and artefacts that would not be present in the simulated data. After training, the physical model is discarded, and the remaining 3D CNN performs direct inversion from diffraction data to real-space images. For evaluation, simulated and experimental datasets were used and the model performance was measured using metrics such as the χ² error and the structural similarity index (SSIM). The trained network predictions are also used as initial guesses for conventional iterative phase retrieval to improve the speed and accuracy of this method.

AutoPhaseNN demonstrates significant speed improvements compared to traditional iterative phase retrieval methods. Specifically, the model achieves approximately 100x speedup using only the trained CNN portion of the model. The image quality of the direct inversion from AutoPhaseNN is comparable to the results obtained through the 600 iterations of the traditional iterative phase retrieval. The paper provides visual comparisons between the results of the traditional phase retrieval, the direct inversion from AutoPhaseNN and the refinement method on both simulated and experimental data using volume rendering and quantitative measures, such as χ² error and SSIM. When used as a prior for iterative phase retrieval, the AutoPhaseNN predictions further enhance the speed of reconstruction by a factor of 10, achieving comparable or even better image quality than iterative methods alone, even after a limited number of iterations of 50 (as opposed to 600). Histograms of χ² errors for the diffraction patterns and SSIM values for real-space images show the effectiveness of the proposed approach. The paper addresses concerns about overfitting by evaluating a free R-factor, demonstrating that the combined approach of AutoPhaseNN prediction followed by refinement yields the best performance.

The results show that AutoPhaseNN successfully addresses the limitations of both traditional iterative phase retrieval and supervised deep learning methods. The unsupervised nature of the training process eliminates the need for large, computationally expensive labeled datasets, making it practical for real-world applications. The speed improvements offer significant advantages for real-time coherent diffraction imaging, particularly with the advent of high-throughput experimental setups. The combined approach, using AutoPhaseNN predictions as priors for iterative refinement, provides an optimal balance between speed and image quality. This methodology has broad implications for a wide range of phase retrieval problems, whenever the forward model is well known.

AutoPhaseNN presents a novel unsupervised deep learning approach for phase retrieval in 3D BCDI, offering significant speed improvements (approximately 100x faster than traditional methods) and maintaining comparable image quality. The unsupervised nature of the training eliminates the need for labeled data, and using its prediction as a learned prior further improves iterative phase retrieval. This work opens new avenues for real-time, high-throughput coherent diffraction imaging and has broader applicability to various phase retrieval and inverse problems.

The current implementation of AutoPhaseNN is primarily trained on simulated data from fcc gold crystals, and a small number of experimental datasets. While the transfer learning strategy using fine-tuning on experimental data has been demonstrated to be effective, a more diverse and extensive training dataset, including crystals of different space groups and with defects, is expected to improve its performance and generalization capabilities to various materials and experimental setups. The applicability to samples beyond the ones considered in this study requires further investigation.