Solving Differential Equations with Neural Networks: Application to the Normal-mode Equation of Sound Field under the Condition of Ideal Shallow Water Waveguide

This paper presents an approach for improving the stability and adaptability of normal-mode theory, applied to marine environment sound field modeling and calculation. Firstly, based on the custom loss function, both Single Output Joint Neural Network (SOJNN) and Multi Output Neural Network (MONN) with physical constraints are constructed to obtain the prediction of wave numbers and eigenfunction. Secondly, we converted the complex wave numbers of normal-mode into a function to find the extreme value, as well as combining the boundary conditions set as the loss function. In this way, the problem space is constrained to the solution space and the dimension of the search space is reduced. Finally, the corresponding sound field is obtained according to the wave numbers and eigenfunction output by neural network. We set the sound velocity at 1500m/s with the density of water is 1g/cm
³
, place the sound source at 32 meters underwater. The upper boundary is absolutely soft and the lower boundary is absolutely hard. The effectiveness of our method is verified by simulation.