A novel multi-modal neurosymbolic reasoning intelligent algorithm for BLMP equation

The (3+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation serves as a crucial nonlinear evolution equation in mathematical physics, capable of characterizing complex nonlinear dynamic phenomena in three-dimensional space and one-dimensional time. With broad applications spanning fluid dynamics, shallow water waves, plasma physics, and condensed matter physics, the investigation of its solutions holds significant importance. Traditional analytical methods face limitations due to their dependence on bilinear forms. To overcome this constraint, this letter proposes a novel multi-modal neurosymbolic reasoning intelligent algorithm (MMNRIA) that achieves 100% accurate solutions for nonlinear partial differential equations without requiring bilinear transformations. By synergistically integrating neural networks with symbolic computation, this approach establishes a new paradigm for universal analytical solutions of nonlinear PDEs. As a practical demonstration, we successfully derive several exact analytical solutions for the (3+1)-dimensional BLMP equation using MMNRIA. These solutions provide a powerful theoretical framework for studying intricate wave phenomena governed by nonlinearity and dispersion effects in three-dimensional physical space.