Eikonal tomography, or travel time inversion, has been one of the primary seismological tools for decades and has been used to understand Earth’s properties and dynamic processes. At the heart of the inversion process is the need for an accurate, and preferably flexible, eikonal solver to compute the travel time field. Most of the conventional eikonal solvers, however, suffer from first-order convergence errors and difficulties in dealing with irregular computational grids. Physics-informed neural networks (PINNs) have been introduced to tackle these problems and have shown great success in addressing those challenges. Nevertheless, these approaches still suffer from slow convergence and unstable training dynamics due to the multi-term nature of the loss function. To improve on this, we propose a new formulation for the isotropic eikonal equation, which imposes boundary conditions as hard constraints. We employ the theory of functional connections to the eikonal tomography problem, which allows for the utilization of a single loss term for training the PINN model. Through rigorous numerical tests, its efficiency, stability, and flexibility in tackling a variety of cases, including topography-dependent and 3D models, are attested, thus providing an efficient and stable PINN-based eikonal tomography.

Full waveform inversion (FWI), a state-of-the-art seismic inversion algorithm, comprises an iterative data-fitting process to recover high-resolution Earth’s properties (e.g., velocity). At the heart of this process lies the numerical wave equation solver, which necessitates discretization. To perform efficient discretization-free FWI for large-scale problems, we introduce physics-informed neural networks (PINNs) as surrogates for conventional numerical solvers. Unfortunately, the original PINN implementation requires additional training for the new velocity model when used in the forward simulation. To make PINNs more suitable for such scenarios, we introduce latent representation learning to PINNs. We first append the input with the encoded velocity vectors, which are the latent representation of the velocity models using an autoencoder model. Unlike the original implementation, the trained PINN model can instantly produce different wavefield solutions without retraining with this additional information. To further improve the FWI efficiency, instead of computing the FWI updates on the original velocity dimension, we resort to updating in its latent representation dimension. Specifically, only the latent representation vectors get updated while the weights of the autoencoder and the PINN model are kept fixed during FWI. Through a series of numerical tests, the proposed framework shows a significant increase in accuracy and computational efficiency compared to the conventional FWI. The improved performance of our framework can be attributed to implicit regularization introduced by the velocity encoding and physics-informed training procedures. The proposed framework presents a significant step forward in utilizing a discretization-free wave equation solver for a more efficient and accurate FWI application.