Stable neural network-based traveltime tomography using hard-constrained measurements

Traveltime tomography, or traveltime inversion, has been one of the primary seismological tools for decades and has been used to understand the earth’s properties and dynamic processes. An accurate, preferably flexible, eikonal solver to compute the traveltime field is at the heart of the inversion process. However, most conventional eikonal solvers suffer from first-order convergence errors and difficulties dealing with irregular computational grids. Physics-informed neural networks (PINNs) have been introduced to tackle these problems and have successfully addressed these challenges. Nevertheless, these approaches still suffer from slow convergence and unstable training dynamics due to the multiterm nature of the loss function. To improve this, we develop a new formulation for the isotropic eikonal equation, which imposes boundary conditions as hard constraints. We apply the theory of functional connections to the traveltime tomography problem, which allows for using a single loss term to train the PINN model. We also analyze the effect of different traveltime factorizations on the overall inversion performance. The additive factorization yields a better result than the previously used multiplicative factorization. Our framework’s efficiency, stability, and flexibility in tackling various cases, such as topography-dependent and 3D models, are tested through rigorous numerical tests, thus providing an efficient and stable PINN-based traveltime tomography. Compared with existing PINN-based inversion, our framework introduces more stability during the inversion and offers significant convergence speedups.