Physics-Informed Fourier-DeepONet for a generalized eikonal solution

The accurate calculation of seismic traveltime based on the eikonal equation has numerous applications in geophysics, such as microseismic localization and tomography. With the advancement of deep learning, the emergence of neural operators has enabled neural networks to learn general solutions to partial differential equations (PDEs). Moreover, Physics-Informed Neural Network (PINN) allows deep learning models to learn under the supervision of PDEs rather than relying solely on training labels. In this context, we propose utilizing a hybrid model that combines the Deep Operator Network (DeepONet) with the Fourier Neural Operator (FNO) to simulate seismic traveltime under the guidance of eikonal equation, thereby yielding a general solution. We refer to this approach as the Physics-Informed Fourier-DeepONet (PI-Fourier-DeepONet). The loss function of the eikonal equation is calculated by finite difference scheme. We evaluate this method across four different types of seismic structures, and the results demonstrate that PI-Fourier-DeepONet is applicable to a wide range of complex geological structures.