Deep-learning optimization using the gradient of a custom objective function: A full-waveform inversion example study on the convolutional objective function

The integration of conventional high-performance full-waveform inversion (FWI) algorithms with deep-learning frameworks is an innovative and promising research direction with the potential to enhance and broaden the application prospects of this field significantly. Automatic differentiation with backpropagation techniques can derive the gradients of model parameters in an equivalent manner to that of adjoint methods; however, it requires a substantial amount of computer memory, particularly when considering the time-step layers. In addition, some excellent objective functions suitable for FWI are not available in deep-learning frameworks. In comparison, the adjoint method, based on effective boundary storage technology, offers greater practicality for calculating the gradients of various objective functions. Therefore, this paper develops a novel approach toward FWI that integrates deep-learning optimization with high-performance gradient computation. In particular, our method inputs model parameter gradients from a custom objective function into the deep-learning framework. Herein, we use the acoustic equation with variable density as an example to demonstrate how a convolutional objective function, along with its corresponding velocity and density gradients, can be used for optimized inversion, multiscale inversion, and deep network parameterization-based multiscale inversion within the deep-learning framework. This approach provides a paradigm for deep-learning-optimized FWI, which we apply to synthetic and field data scenarios.