Physics-informed neural wavefields with Gabor basis functions

Recently, Physics-Informed Neural Networks (PINNs) have gained significant attention for their versatile interpolation capabilities in solving partial differential equations (PDEs). Despite their potential, the training can be computationally demanding, especially for intricate functions like wavefields. This is primarily due to the neural-based (learned) basis functions, biased toward low frequencies, as they are dominated by polynomial calculations, which are not inherently wavefield-friendly. In response, we propose an approach to enhance the efficiency and accuracy of neural network wavefield solutions by modeling them as linear combinations of Gabor basis functions that satisfy the wave equation. Specifically, for the Helmholtz equation, we augment the fully connected neural network model with an adaptable Gabor layer constituting the final hidden layer, employing a weighted summation of these Gabor neurons to compute the predictions (output). These weights/coefficients of the Gabor functions are learned from the previous hidden layers that include nonlinear activation functions. To ensure the Gabor layer's utilization across the model space, we incorporate a smaller auxiliary network to forecast the center of each Gabor function based on input coordinates. Realistic assessments showcase the efficacy of this novel implementation compared to the vanilla PINN, particularly in scenarios involving high-frequencies and realistic models that are often challenging for PINNs.

Seismic evidence for a 1000 km mantle discontinuity under the Pacific

Seismic discontinuities in the mantle are indicators of its thermo-chemical state and offer clues to its dynamics. Ray-based seismic methods, though limited by the approximations made, have mapped mantle transition zone discontinuities in detail, but have yet to offer definitive conclusions on the presence and nature of mid-mantle discontinuities. Here, we show how to use a wave-equation-based imaging method, reverse-time migration of precursors to surface-reflected seismic body waves, to uncover both mantle transition zone and mid-mantle discontinuities, and interpret their physical nature. We observe a thinned mantle transition zone southeast of Hawaii, and a reduction in impedance contrast around 410 km depth in the same area, suggesting a hotter-than-average mantle in the region. Here, we furthermore reveal a 4000-5000 km-wide reflector in new images of the mid mantle below the central Pacific, at 950-1050 km depth. This deep discontinuity exhibits strong topography and generates reflections with polarity opposite to those originating at the 660 km discontinuity, implying an impedance reversal near 1000 km. We link this mid-mantle discontinuity to the upper reaches of deflected mantle plumes upwelling in the region. Reverse-time migration full-waveform imaging is a powerful approach to imaging Earth's interior, capable of broadening our understanding of its structure and dynamics and shrinking modeling uncertainties.

Laplace HypoPINN: Physics-Informed Neural Network for hypocenter localization and its predictive uncertainty

Several techniques have been proposed over the years for automatic hypocenter localization. While those techniques have pros and cons that trade-off computational efficiency and the susceptibility of getting trapped in local minima, an alternate approach is needed that allows robust localization performance and holds the potential to make the elusive goal of real-time microseismic monitoring possible. Physics-informed neural networks (PINNs) have appeared on the scene as a flexible and versatile framework for solving partial differential equations (PDEs) along with the associated initial or boundary conditions. We develop HypoPINN -- a PINN-based inversion framework for hypocenter localization and introduce an approximate Bayesian framework for estimating its predictive uncertainties. This work focuses on predicting the hypocenter locations using HypoPINN and investigates the propagation of uncertainties from the random realizations of HypoPINN's weights and biases using the Laplace approximation. We train HypoPINN to obtain the optimized weights for predicting hypocenter location. Next, we approximate the covariance matrix at the optimized HypoPINN's weights for posterior sampling with the Laplace approximation. The posterior samples represent various realizations of HypoPINN's weights. Finally, we predict the locations of the hypocenter associated with those weights' realizations to investigate the uncertainty propagation that comes from those realisations. We demonstrate the features of this methodology through several numerical examples, including using the Otway velocity model based on the Otway project in Australia.

Direct domain adaptation through reciprocal linear transformations

We propose a direct domain adaptation (DDA) approach to enrich the training of supervised neural networks on synthetic data by features from real-world data. The process involves a series of linear operations on the input features to the NN model, whether they are from the source or target distributions, as follows: (1) A cross-correlation of the input data (i.e., images) with a randomly picked sample pixel (or pixels) of all images from the input or the mean of all randomly picked sample pixel (or pixels) of all input images. (2) The convolution of the resulting data with the mean of the autocorrelated input images from the other domain. In the training stage, as expected, the input images are from the source distribution, and the mean of auto-correlated images are evaluated from the target distribution. In the inference/application stage, the input images are from the target distribution, and the mean of auto-correlated images are evaluated from the source distribution. The proposed method only manipulates the data from the source and target domains and does not explicitly interfere with the training workflow and network architecture. An application that includes training a convolutional neural network on the MNIST dataset and testing the network on the MNIST-M dataset achieves a 70% accuracy on the test data. A principal component analysis (PCA), as well as t-SNE, shows that the input features from the source and target domains, after the proposed direct transformations, share similar properties along the principal components as compared to the original MNIST and MNIST-M input features.