Abundant exact solutions of the fractional (3+1)-dimensional Yu–Toda–Sasa–Fukuyama (YTSF) Equation using the Bell Polynomial-based neural network method

In this paper, we propose a novel neural network method based on Bell polynomials to solve the fractional (3+1)-dimensional Yu–Toda–Sasa–Fukuyama (YTSF) equation, which describes wave phenomena in dual-layer liquids and elastic lattices. The bilinear form of this equation is derived using the bell polynomial approach and combined with neural network, which conquers the inconvenience of traditional Bilinear neural network method. Based on this new approach, various neural network architectures were constructed, including “4-2-1,” “4-3-1,” and “4-2-2-1,” incorporating different activation functions to derive exact solutions such as the breather solution, lump solution, the superposition of rogue and lump solutions, as well as the superposition of double periodic lump solutions. Then the dynamic properties of these solutions are illustrated through 3-D plots, density plots, and line plots. In addition, the chaotic behavior of these exact solutions was studied by applying the Duffing chaotic system. More over, N-soliton solutions could also be reconstructed in the proposed neural network framework which has not been previously exhibited and reported to our knowledge. It paves the possible way to seek for new types of interaction between N-soliton solutions and other type solutions in future work.The successful application of the proposed method to solving the fractional (3+1)-dimensional YTSF equation demonstrates its effectiveness and shows its application potential in fields such as fluid dynamics, marine engineering, and meteorological modeling.