Small-data-driven fast seismic simulations for complex media using physics-informed Fourier neural operators

Abstract

Deep learning (DL) seismic simulations have become a leading-edge field that could provide an effective alternative to traditional numerical solvers. We have developed a small-data-driven time-domain method for fast seismic simulations in complex media based on the physics-informed Fourier neural operator (FNO). Unlike most DL-based modeling schemes that either solve wave equations by embedding the physical constraints into the cost function or conduct physics-informed learning by incorporating the wave functions into convolutional neural networks (CNNs), the FNO uses a learning architecture similar to the structure of split-step Fourier wave propagators, which is composed of two CNNs formulated in the space and wavenumber domains, respectively. The space-domain CNN acts as a local trainable phase-screen compensation. The wavenumber-domain CNN represents a nonlocal spatial convolutional operator acting as a trainable wavenumber filter for the phase-shift process. The FNO method approximates the mathematical-physical behavior of wave equations through learning the mapping between seismic wavefields at different time/locations from training seismic data. That is, the learning process parameterizes the integral kernel directly in the Fourier space, so that we can establish an expressive and efficient architecture for a better balance between accuracy and performance than the traditional spatial CNNs. Applications to gradient, layered, and Marmousi velocity models demonstrate its performance in accuracy and efficiency. The FNO seismic simulation is a data-driven method that needs a small amount of training data, especially when using blended source training data. It is a discretization-independent method that is not subject to the limitation of spatial sampling and time steps imposed on traditional numerical solvers, implying that the training data can be discretized arbitrarily. It also is a model-independent method that can include absorption attenuation into seismic modeling without the need of viscoelastic wave equations.