Abstract: Applications of neural network algorithms in rock physics have developed rapidly developed, mainly due to the neural network's powerful abilities in data modeling, signal processing, and image recognition. However, mathematical and physical explanations of neural networks remain limited, which makes it difficult to understand the behavior and mechanism of neural networks and limits their further development. Using mathematical and physical methods to explain the behavior of neural networks remains a challenging task. The goal of this study was to design a sound wave neural network (SWNN) structure based on sound wave partial differential equations and finite difference methods. The method transforms the first-order sound equations into the frequency domain and discretizes them using a central difference scheme. The differential formula takes the same form as the propagation function of a neural network, enabling the construction of a sound wave neural network. The main features of the SWNN are (1) a neural network with explicitly coupled pressure-velocity streams and inter-layer connections and (2) an adjoint variable method to improve the vanishing gradient problem in network training. The sound wave neural network established from the sound wave partial differential equation and finite difference algorithm has a solid mathematical modeling process and a clear physical explanation. This makes improving network performance within the framework of the mathematical and physical methods feasible. The numerical results showed that SWNN outperforms residual neural networks in image classification on CIFAR-10 and CIFAR-100 datasets. The partial differential equation neural network modeling method proposed in this paper can be applied to many other types of mathematical physics equations, providing a deep mathematical explanation for neural networks.