Modal wavenumber estimation by combining physical informed neural network

Estimation of modal wavenumbers is important for inference of geoacoustic properties and data-driven matched field processing in shallow water waveguides. This paper introduces a deep neural network called combining physical informed neural network (CPINN) for modal wavenumber estimation using a vertical linear array (VLA). Note that the sound field recorded by a VLA can be expressed as a linear superposition of finite modal depth functions, and the differential equations satisfied by the modal depth functions are related to the corresponding modal wavenumbers. CPINN can estimate the modal wavenumbers by introducing the proxies of the modal depth functions and constraining them to satisfy the corresponding differential equations. The performance of the CPINN is evaluated by simulated data in a noisy shallow water environment. Numerical results show that, when compared with the previous methods, CPINN does not need to know the exact horizontal distance between the sound source and the VLA. Moreover, CPINN can estimate the modal wavenumbers at the VLA position in the case where the range segment traversed by the source, i.e., the aperture in the range direction, is smaller than the maximum modal cycle distance and in a range-dependent waveguide.