In this article, we propose physics-informed neural operators (PINO) that combine training data and physics constraints to learn the solution operator of a given family of parametric Partial Differential Equations (PDE). PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. Specifically, in PINO, we combine coarse-resolution training data with PDE constraints imposed at a higher resolution. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families and shows no degradation in accuracy even under zero-shot super-resolution, that is, being able to predict beyond the resolution of training data. PINO uses the Fourier neural operator (FNO) framework that is guaranteed to be a universal approximator for any continuous operator and discretization convergent in the limit of mesh refinement. By adding PDE constraints to FNO at a higher resolution, we obtain a high-fidelity reconstruction of the ground-truth operator. Moreover, PINO succeeds in settings where no training data is available and only PDE constraints are imposed, while previous approaches, such as the Physics-Informed Neural Network (PINN), fail due to optimization challenges, for example, in multi-scale dynamic systems such as Kolmogorov flows.

Seismic wave propagation forms the basis for most aspects of seismological research, yet solving the wave equation is a major computational burden that inhibits the progress of research. This is exacerbated by the fact that new simulations must be performed whenever the velocity structure or source location is perturbed. Here, we explore a prototype framework for learning general solutions using a recently developed machine learning paradigm called neural operator. A trained neural operator can compute a solution in negligible time for any velocity structure or source location. We develop a scheme to train neural operators on an ensemble of simulations performed with random velocity models and source locations. As neural operators are grid free, it is possible to evaluate solutions on higher resolution velocity models than trained on, providing additional computational efficiency. We illustrate the method with the 2D acoustic wave equation and demonstrate the method’s applicability to seismic tomography, using reverse-mode automatic differentiation to compute gradients of the wavefield with respect to the velocity structure. The developed procedure is nearly an order of magnitude faster than using conventional numerical methods for full waveform inversion.

Seismic waveform modeling is a powerful tool for determining earth structure models and unraveling earthquake rupture processes, but it is usually computationally expensive. We introduce a scheme to vastly accelerate these calculations with a recently developed machine learning paradigm called the neural operator. Once trained, these models can simulate a full wavefield at negligible cost. We use a U-shaped neural operator to learn a general solution operator to the 2D elastic wave equation from an ensemble of numerical simulations performed with random velocity models and source locations. We show that full waveform modeling with neural operators is nearly two orders of magnitude faster than conventional numerical methods, and more importantly, the trained model enables accurate simulation for velocity models, source locations, and mesh discretization distinctly different from the training dataset. The method also enables convenient full-waveform inversion with automatic differentiation.

Seismic wave arrival time measurements form the basis for numerous downstream applications. State‐of‐the‐art approaches for phase picking use deep neural networks to annotate seismograms at each station independently, yet human experts annotate seismic data by examining the whole network jointly. Here, we introduce a general‐purpose network‐wide phase picking algorithm based on a recently developed machine learning paradigm called Neural Operator. Our model, called Phase Neural Operator, leverages the spatio‐temporal contextual information to pick phases simultaneously for any seismic network geometry. This results in superior performance over leading baseline algorithms by detecting many more earthquakes, picking more phase arrivals, while also greatly improving measurement accuracy. Following similar trends being seen across the domains of artificial intelligence, our approach provides but a glimpse of the potential gains from fully‐utilizing the massive seismic data sets being collected worldwide.

We present a data‐driven framework for ground‐motion synthesis that generates three‐component acceleration time histories conditioned on moment magnitude (M), rupture distance (⁠ $Rrup$⁠), time‐average shear‐wave velocity at the top 30 m (⁠ $VS30$⁠), and style of faulting. We use a Generative Adversarial Neural Operator (GANO)—a resolution invariant architecture that guarantees model training independent of the data sampling frequency. We first present the conditional ground‐motion synthesis algorithm (cGM‐GANO) and discuss its advantages compared to the previous work. We next train cGM‐GANO on simulated ground motions generated by the Southern California Earthquake Center Broadband Platform (BBP) and on recorded the Kiban–Kyoshin network (KiK‐net) data, and show that the model can learn the overall magnitude, distance, and $VS30$ scaling of effective amplitude spectra (EAS) ordinates and pseudospectral accelerations (PSA). Results specifically show that cGM‐GANO produces consistent median scaling with the training data for the corresponding tectonic environments over a wide range of frequencies for scenarios with sufficient data coverage. For the BBP dataset, cGM‐GANO cannot learn the ground‐motion scaling of the stochastic frequency components (f > 1 Hz); for the KiK‐net dataset, the largest misfit is observed at short distances (⁠ $Rrup<50km$⁠) and for soft‐soil conditions (⁠ $VS30<200m/s$⁠) due to the scarcity of such data. Except for these conditions, the aleatory variability of EAS and PSA are captured reasonably well. Finally, cGM‐GANO produces similar median scaling to traditional ground‐motion models (GMMs) for frequencies greater than 1 Hz for both PSA and EAS but underestimates the aleatory variability of EAS. Discrepancies in the comparisons between the synthetic ground motions and GMMs are attributed to inconsistencies between the training dataset and the datasets used in GMM development. Our pilot study demonstrates GANO’s potential for efficient synthesis of broadband ground motions.