In this paper, we discuss an unsupervised deep learning (DL) method for solving time domain electromagnetic simulations. Compared to the conventional approach, our method encodes initial conditions, boundary conditions as well as Maxwell's equations as the constraints when training the network, turning an electromagnetic simulation problem into an optimization process. High prediction accuracy of the electromagnetic fields, without discretization or interpolation in space or in time, can be achieved with limited number of layers and neurons in each layer of the neural network. We study several numerical examples to demonstrate the effectiveness of this method for simulating time-domain electromagnetic fields. First, the accuracy of this method is validated by comparing with the analytical solution of a 1D cavity model filled with homogeneous media. Then, we combine the continuity condition to modify the loss function for handling medium discontinuities. Further, the computational efficiency of finite-difference and DL methods in conductive and nonlinear media is compared. Finally, we prove the effectiveness of this method in high-dimensional and multi-scale simulations.

In this article, a new scheme based on the supervised descent method (SDM) for solving directional electromagnetic logging-while-drilling (LWD) inverse problems is proposed. The SDM provides us a new perspective to combine the classical gradient-based inversion and machine-learning-based inversion schemes. It iteratively learns a set of descent directions in the offline training process, where the training model set is generated in advance according to the prior information, and then updates the models with the learned descent directions as well as data residuals in the prediction stage, resulting in great flexibility to incorporate prior information, the capability of skipping local minima, and accelerated convergence. The generalization ability of the SDM to interrogate new models that are not contained in the training model set is also explored. By utilizing real-time information obtained from the logging process, the learned descent directions can be slightly revised with a higher efficiency to get closer to the true model. In addition, we probe the sensitivity of the SDM by adding different levels of random noise to the measurements. Numerical examples demonstrate that SDM-based inversion can achieve a higher resolution, faster convergence, and higher robustness than conventional schemes such as Occam's inversion.

A deep-learning-based workflow is proposed in this paper to solve the first-arrival picking problem for near-surface velocity model building. Traditional methods, such as the short-term average/long-term average method, perform poorly when the signal-to-noise ratio is low or near-surface geologic structures are complex. This challenging task is formulated as a segmentation problem accompanied by a novel postprocessing approach to identify pickings along the segmentation boundary. The workflow includes three parts: a deep U-net for segmentation, a recurrent neural network (RNN) for picking, and a weight adaptation approach to be generalized for new data sets. In particular, we have evaluated the importance of selecting a proper loss function for training the network. Instead of taking an end-to-end approach to solve the picking problem, we emphasize the performance gain obtained by using an RNN to optimize the picking. Finally, we adopt a simple transfer learning scheme and test its robustness via a weight adaptation approach to maintain the picking performance on new data sets. Our tests on synthetic data sets reveal the advantage of our workflow compared with existing deep-learning methods that focus only on segmentation performance. Our tests on field data sets illustrate that a good postprocessing picking step is essential for correcting the segmentation errors and that the overall workflow is efficient in minimizing human interventions for the first-arrival picking task.

To effectively overcome the cycle-skipping issue in full-waveform inversion (FWI), we have developed a deep neural network (DNN) approach to predict the absent low-frequency (LF) components by exploiting the hidden physical relation connecting the LF and high-frequency (HF) data. To efficiently solve this challenging nonlinear regression problem, two novel strategies are proposed to design the DNN architecture and to optimize the learning process: (1) the dual data feed structure and (2) progressive transfer learning. With the dual data feed structure, not only the HF data, but also the corresponding beat tone data, are fed into the DNN to relieve the burden of feature extraction. The second strategy, progressive transfer learning, enables us to train the DNN using a single evolving training data set. Within the framework of progressive transfer learning, the training data set continuously evolves in an iterative manner by gradually retrieving the subsurface information through the physics-based inversion module, progressively enhancing the prediction accuracy of the DNN and propelling the inversion process out of the local minima. The synthetic numerical experiments suggest that, without any a priori geologic information, the LF data predicted by the progressive transfer learning are sufficiently accurate for an FWI engine to produce reliable subsurface velocity models free of cycle-skipping artifacts.

In this communication, a trainable theory-guided recurrent neural network (RNN) equivalent to the finite-difference-time-domain (FDTD) method is exploited to formulate electromagnetic propagation, solve Maxwell’s equations, and the inverse problem on differentiable programming platform Pytorch. For forward modeling, the computation efficiency is substantially improved compared to conventional FDTD implemented on MATLAB. Gradient computation becomes more precise and faster than the traditional finite difference method benefiting from the accurate and efficient automatic differentiation on the differentiable programming platform. Moreover, by setting the trainable weights of RNN as the material-related parameters, an inverse problem can be solved by training the network. Numerical results demonstrate the effectiveness and efficiency of the method for forward and inverse electromagnetic modeling.

Joint inversion has drawn considerable attention due to the availability of multiple geophysical data sets, ever-increasing computational resources, the development of advanced algorithms, and its ability to reduce inversion uncertainty. A key issue of joint inversion is to develop effective strategies to link different geophysical data in a unified mathematical framework, in which the information obtained from different models can complement each other. We have developed a deep learning-enhanced joint inversion framework to simultaneously reconstruct different physical models by fusing different types of geophysical data. Traditionally, structure similarity constraints are pursued by joint inversion algorithms using manually crafted formulations (e.g., cross gradient). The constraint is constructed by a deep neural network (DNN) during the learning process. The framework is designed to combine the DNN and a traditional independent inversion workflow and improve the joint inversion result iteratively. The network can be easily extended to incorporate multiphysics without structural changes. Numerical experiments on the joint inversion of 2D DC resistivity data and seismic traveltime are used to validate our method. In addition, this learning-based framework demonstrates excellent generalization abilities when tested on data sets using different geologic structures. It also can handle different sensing configurations and nonconforming discretization.