This paper discuss fixed-point theories can be used to train deep neural networks. We find a mathematical link between nonlinear analysis and learning dynamics by looking at neural network optimisation as a fixed-point problem. In repeated training processes, fixed-point formulas ensure convergence, stability, and equilibrium in a theoretical sense.  The results of experiments show that fixed-point–based algorithms are faster and more stable than standard optimisation methods such as Adam and gradient descent.  This study shows how fixed-point principles can help make learning methods for complicated neural systems more accurate, easy to understand, and effective.