Seismic travel-time tomography is an effective approach for exploring subsurface structures. In recent years, machine learning (ML) has been successfully applied in seismic travel-time tomography; however, the uncertainty quantification of predictive outputs has been less frequently addressed. To tackle these challenges, we adopt the deep ensemble of deep operator networks with randomized priors (DERP-DON) for travel-time tomography, based on measured travel-time data from surface and cross-well measurements. Deep ensembles provide informative uncertainty estimates, and the incorporation of randomized priors allows the network model to offer more conservative uncertainty quantification. Through numerical experiments, we demonstrate that DERP-DON can provide reasonable and conservative uncertainty estimates while accurately inferring the velocity field. Notably, DERP-DON is particularly effective in offering meaningful uncertainty estimates for out-of-distribution data.

Seismic travel time plays a fundamental role in a wide array of geophysical applications. Traditionally, numerical simulation of travel time involves solving the eikonal equation. However, conventional methods are typically limited to simulating the travel-time field for a single source and velocity model at a time. This limitation poses challenges, particularly when dealing with inverse problems that necessitate multiple forward simulations to infer velocity models based on travel-time data excited by different sources. In recent years, machine learning has proven its effectiveness in tackling problems associated with partial differential equations (PDEs). Among these methods, the Deep Operator Network (DeepONet) has gained attention for its adaptable structure and minimal generalization error. In response to the challenges posed by solving the eikonal equation in heterogeneous media, we introduce a modified architecture known as the Fully Convolutional DeepONet (FC-DeepONet). This approach leverages convolutional operations to extract features directly from 2D data and avoid flattening operations that could lead to the loss of important spatial information. The FC-DeepONet model takes the velocity model and source location as input and generates the corresponding travel-time fields as output. Through numerical experiments, we validate the efficacy of our proposed method in accurately predicting travel-time fields induced by sources located at various positions across diverse velocity models. Besides, our approach demonstrates robustness by providing reasonably accurate predictions even in scenarios involving velocity models with irregular topography. This adaptability holds significant promise for practical applications, particularly in cases characterized by complex geological features.

The high cost of acquiring a sufficient amount of seismic data for training has limited the use of machine learning in seismic tomography. In addition, the inversion uncertainty due to the noisy data and data scarcity is less discussed in conventional seismic tomography literature. To mitigate the uncertainty effects and quantify their impacts in the prediction, the so-called Bayesian Physics-Informed Neural Networks (BPINNs) based on the eikonal equation are adopted to infer the velocity field and reconstruct the travel-time field. In BPINNs, two inference algorithms including Stein Variational Gradient Descent (SVGD) and Gaussian variational inference (VI) are investigated for the inference task. The numerical results of several benchmark problems demonstrate that the velocity field can be estimated accurately and the travel-time can be well approximated with reasonable uncertainty estimates by BPINNs. This suggests that the inferred velocity model provided by BPINNs may serve as a valid initial model for seismic inversion and migration.

Seismic inversion is a significant tool for exploring the structure and characteristics of the underground. However, the conventional inversion strategy strongly depends on the initial model. In this work, we employ the physics-informed neural network (PINN) to estimate the velocity and density fields based on acoustic wave equations. In contrast to the traditional purely data-driven machine learning approaches, PINNs leverage both available data and the physical laws that govern the observed data during the training stage. In this work, the first-order acoustic wave equations are embedded in the loss function as a regularization term for training the neural networks. In addition to the limited amount of measurements about the state variables available at the surface being used as the observational data, the well logging data is also used as the direct observational data about the model parameters. The numerical results from several benchmark problems demonstrate that given noise-free or noisy data, the proposed inversion strategy is not only capable of predicting the seismograms, but also estimating the velocity and density fields accurately. Finally, we remark that although the absorbing boundary conditions are not imposed in the proposed method, the reflected waves do not appear from the artificial boundary in the predicted seismograms.

In this work, we predicted ankle joint torque by combining a neuromusculoskeletal (NMS) solver-informed artificial neural network (hybrid-ANN) model with transfer learning based on joint angle and muscle electromyography signals. The hybrid-ANN is an ANN augmented with two kinds of features: 1) experimental measurements - muscle signals and joint angles, and 2) informative physical features extracted from the underlying NMS solver, such as individual muscle force and joint torque. The hybrid-ANN model accuracy in torque prediction was studied in both intra- and inter-subject tests, and compared to the baseline models (NMS and standard-ANN). For each prediction model, seven different cases were studied using data from gait at different speeds and from isokinetic ankle dorsi/plantarflexion motion. Additionally, we integrated a transfer learning method in inter-subject models to improve joint torque prediction accuracy by transferring the learned knowledge from previous participants to a new participant, which could be useful when training data is limited. Our results indicated that better accuracy could be obtained by integrating informative NMS features into a standard ANN model, especially in inter-subject cases; overall, the hybrid-ANN model predicted joint torque with higher accuracy than the baseline models, most notably in inter-subject prediction after adopting the transfer learning technique. We demonstrated the potential of combining physics-based NMS and standard-ANN models with a transfer learning technique in different prediction scenarios. This procedure holds great promise in applications such as assistance-as-needed exoskeleton control strategy design by incorporating the physiological joint torque of the users.

In this article, we present one numerical approach to infer the model parameters and state variables of acoustic wave equations. The method we consider is based on the recently proposed method-the so-called hidden physics model. With placing Gaussian process (GP) prior on the state variables, the structure and model parameters of acoustic wave equations are encoded into the kernel function of a multioutput GP. The purpose of this article includes: 1) testing the applicability of hidden physics model to infer the velocity, density, and state variables of the acoustic wave equation, which is important for many applications in geophysics; 2) adapting the method to handle for both homogeneous and heterogeneous media that are of practical interest; and 3) exploring efficient sequential sampling methods to improve the sampling efficiency. We suggest that the expected-improvement-based sequential sampling method would be effective for most practical problems related to acoustic wave propagation. Besides, we demonstrate the performance of the proposed scheme via several benchmark problems.