Exploring the exact solutions to the nonlinear systems with neural networks method.

This paper explores the use of Riccati subequation neural networks to solve nonlinear partial differential equations modeling complex biological processes like cancer, brain function, and wound healing. These equations involve spatially varying biochemical signaling and tissue regeneration. The study applies a method where solutions to the Riccati problem are integrated into neural networks, with the first hidden layer neurons specifically providing the solution to the Riccati equation. Exact solutions for differential equations may be obtained using the suggested approach. In order to verify the mathematical foundation of this technique, we examine the proposed equations, which leads to the derivation of hyperbolic function solutions, trigonometric function solutions, and rational solutions. Using various graphical illustrations, the dynamic properties of certain solutions associated with waves have been demonstrated. The results provided in this study have the potential to improve understanding of the nonlinear dynamic characteristics displayed by the specified system and to confirm the effectiveness of the techniques that have been implemented.