Training a deep operator network as a surrogate solver for two-dimensional parabolic-equation models

Parabolic equations (PEs) are useful for modeling sound propagation in a range-dependent environment. However, this approach entails approximating a leading-order cross-derivative term in the PE square-root operators. Deep operator networks (DeepONets) are designed to approximate operators. In this paper, we train DeepONets to take complex sound pressure and speed of sound at any depth location of interest as inputs and approximate the PE square operator in modeling two-dimensional sound propagation. Once trained, a network can predict the far field for a wide variety of environmental conditions, without needing to approximate the operator or calculate the whole mode trajectory and at a lower computational cost. The original DeepONet learns the operator of a single function. By contrast, the modified version presented here learns multiple-input operators with Fourier features. Using computational and theoretical examples, we demonstrate that DeepONet is efficient for learning complex ocean acoustic physics with good accuracy.