A New Neural Network Framework Integrating Symbolic Computation to Solve the (2+1)-Dimensional Boussinesq Equation

The (2+1)-dimensional Boussinesq equation is a fundamental model in nonlinear wave theory, governing shallow-water wave propagation, coastal dynamics in ocean engineering, and long waves in geophysical fluid systems such as atmospheric and oceanic currents. We present a novel neural network symbolic computation framework that seamlessly integrates neural architectures for powerful function approximation with symbolic manipulation for exact algebraic resolution, eliminating the need for bilinear transformations and thereby substantially reducing computational complexity. Applying this framework, we derive five previously unreported exact analytical solutions using carefully designed neural network configurations and probe functions. These solutions provide valuable tools for modeling ocean internal waves, coastal engineering simulations, and nonlinear optical pulse dynamics. In practice, the method delivers faster and more accurate simulations, improving engineering design and environmental prediction capabilities. By synergistically combining neural networks with symbolic computation, our approach surpasses traditional numerical methods and physics-informed neural networks in both accuracy and efficiency, opening new avenues for solving complex nonlinear partial differential equations.