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Numerical simulations of seismic wave propagation in heterogeneous 3D mediaare central to investigating subsurface structures and understanding earthquakeprocesses, yet are computationally expensive for large problems. This isparticularly problematic for full waveform inversion, which typically involvesnumerous runs of the forward process. In machine learning there has beenconsiderable recent work in the area of operator learning, with a new class ofmodels called neural operators allowing for data-driven solutions to partialdifferential equations. Recent works in seismology have shown that when neuraloperators are adequately trained, they can significantly shorten the computetime for wave propagation. However, the memory required for the 3D time domainequations may be prohibitive. In this study, we show that these limitations canbe overcome by solving the wave equations in the frequency domain, also knownas the Helmholtz equations, since the solutions for a set of frequencies can bedetermined in parallel. The 3D Helmholtz neural operator is 40 times morememory-efficient than an equivalent time-domain version. We employ a U-shapedneural operator for 2D and 3D elastic wave modeling, achieving two orders ofmagnitude acceleration compared to a baseline spectral element method. Theneural operator accurately generalizes to variable velocity structures and canbe evaluated on denser input meshes than used in the training simulations. Wealso show that when solving for wavefields strictly on the surface, theaccuracy can be significantly improved via a graph neural operator layer. Inleveraging automatic differentiation, the proposed method can serve as analternative to the adjoint-state approach for 3D full-waveform inversion,reducing the computation time by a factor of 350.

Deep Neural Helmholtz Operators for 3D Elastic Wave Propagation and Inversion