Learning to solve the elastic wave equation with Fourier neural operators

Abstract

Neural operators are extensions of neural networks, which, through supervised training, learn how to map the complex relationships that exist within the classes of the partial differential equation (PDE). One of these networks, the Fourier neural operator (FNO), has been particularly successful in producing general solutions to PDEs, such as the Navier-Stokes equation. We have formulated an FNO to reproduce solutions of the 2D isotropic elastic wave equation training on synthetic data sets. This requires two significant alterations to the existing FNO structures. By (1) adding the Fourier kernel multiplication with respect to multiple spatial directions and (2) building connections between the Fourier layers, we produce what we refer to as the “one-connection FNO,” which is suitable for use in producing solutions of the elastic wave equation. Post training, the new FNO is examined for accuracy. Compared with the unmodified original FNO, we observe, in particular, an improved prediction of the fields generated with low source frequency, which is suggestive of immediate applicability in inversion. Once trained, the modified FNO operates at approximately 100 times the speed of traditional finite-difference methods on a CPU; this increase in the computational speed, when used within forward modeling, may have important consequences in simulation-intensive inverse problems, such as those based on the Monte Carlo methods.