Uncertainty quantification of two-phase flow in porous media via coupled-TgNN surrogate model

Uncertainty quantification (UQ) of subsurface two-phase flow usually requires numerous executions of forward simulations under varying conditions. In this work, a novel coupled theory-guided neural network (TgNN) based surrogate model is built to facilitate computation efficiency under the premise of satisfactory accuracy. The core notion of this proposed method is to bridge two separate blocks on top of an overall network. They underlie the TgNN model in a coupled form, which reflects the coupling nature of pressure and water saturation in the two-phase flow equation. The TgNN model not only relies on labeled data, but also incorporates underlying scientific theory and experiential rules (e.g., governing equations, stochastic parameter fields, boundary and initial conditions, well conditions, and expert knowledge) as additional components into the loss function. The performance of the TgNN-based surrogate model for two-phase flow problems is tested by different numbers of labeled data and collocation points, as well as the existence of data noise. The proposed TgNN-based surrogate model offers an effective way to solve the coupled nonlinear two-phase flow problem and demonstrates good accuracy and strong robustness when compared with the purely data-driven surrogate model. By combining the accurate TgNN-based surrogate model with the Monte Carlo method, UQ tasks can be performed at a minimum cost to evaluate statistical quantities. Since the heterogeneity of the random fields strongly impacts the results of the surrogate model, corresponding variance and correlation length are added to the input of the neural network to maintain its predictive capacity. The results show that the TgNN-based surrogate model achieves satisfactory accuracy, stability, and efficiency in UQ problems of subsurface two-phase flow.

Inferring electrochemical performance and parameters of Li-ion batteries based on deep operator networks

The Li-ion battery is a complex physicochemical system that generally takes applied current as input and terminal voltage as output. The mappings from current to voltage can be described by several kinds of models, such as accurate but inefficient physics-based models, and efficient but sometimes inaccurate equivalent circuit and black-box models. To realize accuracy and efficiency simultaneously in battery modeling, we propose to build a data-driven surrogate for a battery system while incorporating the underlying physics as constraints. In this work, we innovatively treat the functional mapping from current curve to terminal voltage as a composite of operators, which is approximated by the powerful deep operator network (DeepONet). Its learning capability is firstly verified through a predictive test for Li-ion concentration at two electrodes. In this experiment, the physics-informed DeepONet is found to be more robust than the purely data-driven DeepONet, especially in temporal extrapolation scenarios. A composite surrogate is then constructed for mapping current curve and solid diffusivity to terminal voltage with three operator networks, in which two parallel physics-informed DeepONets are firstly used to predict Li-ion concentration at two electrodes, and then based on their surface values, a DeepONet is built to give terminal voltage predictions. Since the surrogate is differentiable anywhere, it is endowed with the ability to learn from data directly, which was validated by using terminal voltage measurements to estimate input parameters. The proposed surrogate built upon operator networks possesses great potential to be applied in on-board scenarios, such as battery management system, since it integrates efficiency and accuracy by incorporating underlying physics, and also leaves an interface for model refinement through a totally differentiable model structure.

RockGPT: Reconstructing three-dimensional digital rocks from single two-dimensional slice from the perspective of video generation

Random reconstruction of three-dimensional (3D) digital rocks from two-dimensional (2D) slices is crucial for elucidating the microstructure of rocks and its effects on pore-scale flow in terms of numerical modeling, since massive samples are usually required to handle intrinsic uncertainties. Despite remarkable advances achieved by traditional process-based methods, statistical approaches and recently famous deep learning-based models, few works have focused on producing several kinds of rocks with one trained model and allowing the reconstructed samples to satisfy certain given properties, such as porosity. To fill this gap, we propose a new framework, named RockGPT, which is composed of VQ-VAE and conditional GPT, to synthesize 3D samples based on a single 2D slice from the perspective of video generation. The VQ-VAE is utilized to compress high-dimensional input video, i.e., the sequence of continuous rock slices, to discrete latent codes and reconstruct them. In order to obtain diverse reconstructions, the discrete latent codes are modeled using conditional GPT in an autoregressive manner, while incorporating conditional information from a given slice, rock type, and porosity. We conduct two experiments on five kinds of rocks, and the results demonstrate that RockGPT can produce different kinds of rocks with the same model, and the reconstructed samples can successfully meet certain specified porosities. In a broader sense, through leveraging the proposed conditioning scheme, RockGPT constitutes an effective way to build a general model to produce multiple kinds of rocks simultaneously that also satisfy user-defined properties.

Digital rock reconstruction with user-defined properties using conditional generative adversarial networks

Uncertainty is ubiquitous with flow in subsurface rocks because of their inherent heterogeneity and lack of in-situ measurements. To complete uncertainty analysis in a multi-scale manner, it is a prerequisite to provide sufficient rock samples. Even though the advent of digital rock technology offers opportunities to reproduce rocks, it still cannot be utilized to provide massive samples due to its high cost, thus leading to the development of diversified mathematical methods. Among them, two-point statistics (TPS) and multi-point statistics (MPS) are commonly utilized, which feature incorporating low-order and high-order statistical information, respectively. Recently, generative adversarial networks (GANs) are becoming increasingly popular since they can reproduce training images with excellent visual and consequent geologic realism. However, standard GANs can only incorporate information from data, while leaving no interface for user-defined properties, and thus may limit the diversity of reconstructed samples. In this study, we propose conditional GANs for digital rock reconstruction, aiming to reproduce samples not only similar to the real training data, but also satisfying user-specified properties. In fact, the proposed framework can realize the targets of MPS and TPS simultaneously by incorporating high-order information directly from rock images with the GANs scheme, while preserving low-order counterparts through conditioning. We conduct three reconstruction experiments, and the results demonstrate that rock type, rock porosity, and correlation length can be successfully conditioned to affect the reconstructed rock images. Furthermore, in contrast to existing GANs, the proposed conditioning enables learning of multiple rock types simultaneously, and thus invisibly saves the computational cost.

Using Deep Learning to Improve Ensemble Smoother: Applications to Subsurface Characterization

Ensemble smoother (ES) has been widely used in various research fields to reduce the uncertainty of the system-of-interest. However, the commonly-adopted ES method that employs the Kalman formula, that is, ES$_\text{(K)}$, does not perform well when the probability distributions involved are non-Gaussian. To address this issue, we suggest to use deep learning (DL) to derive an alternative update scheme for ES in complex data assimilation applications. Here we show that the DL-based ES method, that is, ES$_\text{(DL)}$, is more general and flexible. In this new update scheme, a high volume of training data are generated from a relatively small-sized ensemble of model parameters and simulation outputs, and possible non-Gaussian features can be preserved in the training data and captured by an adequate DL model. This new variant of ES is tested in two subsurface characterization problems with or without Gaussian assumptions. Results indicate that ES$_\text{(DL)}$ can produce similar (in the Gaussian case) or even better (in the non-Gaussian case) results compared to those from ES$_\text{(K)}$. The success of ES$_\text{(DL)}$ comes from the power of DL in extracting complex (including non-Gaussian) features and learning nonlinear relationships from massive amounts of training data. Although in this work we only apply the ES$_\text{(DL)}$ method in parameter estimation problems, the proposed idea can be conveniently extended to analysis of model structural uncertainty and state estimation in real-time forecasting studies.

Physics-informed semantic inpainting: Application to geostatistical modeling

A fundamental problem in geostatistical modeling is to infer the heterogeneous geological field based on limited measurements and some prior spatial statistics. Semantic inpainting, a technique for image processing using deep generative models, has been recently applied for this purpose, demonstrating its effectiveness in dealing with complex spatial patterns. However, the original semantic inpainting framework incorporates only information from direct measurements, while in geostatistics indirect measurements are often plentiful. To overcome this limitation, here we propose a physics-informed semantic inpainting framework, employing the Wasserstein Generative Adversarial Network with Gradient Penalty (WGAN-GP) and jointly incorporating the direct and indirect measurements by exploiting the underlying physical laws. Our simulation results for a high-dimensional problem with 512 dimensions show that in the new method, the physical conservation laws are satisfied and contribute in enhancing the inpainting performance compared to using only the direct measurements.

Transfer learning for diagnosis of congenital abnormalities of the kidney and urinary tract in children based on Ultrasound imaging data

Classification of ultrasound (US) kidney images for diagnosis of congenital abnormalities of the kidney and urinary tract (CAKUT) in children is a challenging task. It is desirable to improve existing pattern classification models that are built upon conventional image features. In this study, we propose a transfer learning-based method to extract imaging features from US kidney images in order to improve the CAKUT diagnosis in children. Particularly, a pre-trained deep learning model (imagenet-caffe-alex) is adopted for transfer learning-based feature extraction from 3-channel feature maps computed from US images, including original images, gradient features, and distanced transform features. Support vector machine classifiers are then built upon different sets of features, including the transfer learning features, conventional imaging features, and their combination. Experimental results have demonstrated that the combination of transfer learning features and conventional imaging features yielded the best classification performance for distinguishing CAKUT patients from normal controls based on their US kidney images.

Integrating semi-supervised label propagation and random forests for multi-atlas based hippocampus segmentation

A novel multi-atlas based image segmentation method is proposed by integrating a semi-supervised label propagation method and a supervised random forests method in a pattern recognition based label fusion framework. The semi-supervised label propagation method takes into consideration local and global image appearance of images to be segmented and segments the images by propagating reliable segmentation results obtained by the supervised random forests method. Particularly, the random forests method is used to train a regression model based on image patches of atlas images for each voxel of the images to be segmented. The regression model is used to obtain reliable segmentation results to guide the label propagation for the segmentation. The proposed method has been compared with state-of-the-art multi-atlas based image segmentation methods for segmenting the hippocampus in MR images. The experiment results have demonstrated that our method obtained superior segmentation performance.

A dynamic graph-cuts method with integrated multiple feature maps for segmenting kidneys in ultrasound images

Purpose: To improve kidney segmentation in clinical ultrasound (US) images, we develop a new graph cuts based method to segment kidney US images by integrating original image intensity information and texture feature maps extracted using Gabor filters. Methods: To handle large appearance variation within kidney images and improve computational efficiency, we build a graph of image pixels close to kidney boundary instead of building a graph of the whole image. To make the kidney segmentation robust to weak boundaries, we adopt localized regional information to measure similarity between image pixels for computing edge weights to build the graph of image pixels. The localized graph is dynamically updated and the GC based segmentation iteratively progresses until convergence. The proposed method has been evaluated and compared with state of the art image segmentation methods based on clinical kidney US images of 85 subjects. We randomly selected US images of 20 subjects as training data for tuning the parameters, and validated the methods based on US images of the remaining 65 subjects. The segmentation results have been quantitatively analyzed using 3 metrics, including Dice Index, Jaccard Index, and Mean Distance. Results: Experiment results demonstrated that the proposed method obtained segmentation results for bilateral kidneys of 65 subjects with average Dice index of 0.9581, Jaccard index of 0.9204, and Mean Distance of 1.7166, better than other methods under comparison (p<10-19, paired Wilcoxon rank sum tests). Conclusions: The proposed method achieved promising performance for segmenting kidneys in US images, better than segmentation methods that built on any single channel of image information. This method will facilitate extraction of kidney characteristics that may predict important clinical outcomes such progression chronic kidney disease.