Symmetric Fourier Neural Operators With Translation, Rotation, and Scaling Equivariances for Wave Propagation Modeling

Deep-learning description of seismic wave propagation, a physical process that evolves spatiotemporally in complex media, still remains a challenging and ambitious goal that powers many aspects of geophysical applications. Fourier neural operators (FNOs) could provide a potential solution for this goal due to their dual-domain learning architecture, similar to the structure of Fourier wave propagators. The dual-domain FNO architecture does not depend on wave equations and mesh discretizations, enabling small-data-driven fast seismic simulations. It approximates the mathematical-physical behavior of wave equations by learning the mapping between seismic wavefields at different time/locations from training seismic data. However, the wave-propagation-targeted FNOs so far lack the intrinsic symmetric feature of wave equations, such as the equivariances to translation, rotation, and scaling. The trained FNOs are difficult to generalize to different velocity models, frequencies, and source locations in practical applications. In this study, we incorporate the physical symmetries of wave equations into the FNO architecture. The resulting symmetric FNOs (SFNOs) parameterize the convolution kernel directly in the Fourier space through the learning process in terms of the intrinsic invariants of wave equations. The physics-informed SFNOs can be trained to approximate the Green's function of wave equations, enabling the application to arbitrary source wavelets. We propose a new input/output structure to reduce the complexity of seismic wavefields and improve the efficiency of SFNOs. Applications to 2-D and 3-D velocity models demonstrate the performance of SFNOs in accuracy and efficiency across different velocity models, frequencies, and source locations.