Graphical neural networks based on physical information constraints for solving the eikonal equation

Accurate temporal resolution of the eikonal equation forms the cornerstone of seismological studies, including microseismic source localization and traveltime tomography. Physics Informed Neural Networks (PINNs) has gained significant attention as an efficient approximation technique for numerical computations. In this study, we put forth a novel model named Eiko-PIGCNet, a Graph Convolutional Neural Network that incorporates physical constraints. We demonstrate the effectiveness of our proposed model in solving the 3D eikonal equation for travel time estimation. In our approach, the discretized grid points are converted into a graph data structure, where every grid point is regarded as a node, and the neighboring nodes are interconnected via edges. The node characteristics are defined by incorporating the velocity and spatial coordinates of the respective grid points. Ultimately, the efficacy of the Eiko-PIGCNet and PINNs is evaluated and compared under various velocity models. The results reveal that Eiko-PIGCNet outshines PINNs in terms of solution accuracy and computational efficiency.