Modification of high performance training algorithm for solving singular perturbation partial differential equations with cubic convergence

In this study, we used the update a neural network for solve singularly perturbed partial differential equations (SPPDLs) that have been singularly disturbed. We dealt with the modernization of artiftial networks by updating the most important types of high-performance training. For nonlinear equations, the modified Levenberg-Marquardt method (MLMM) is used, in which not only an LM step but also an approximation LM step is generated at each iteration. When applying the trust region technique, a new type of projected reduction for the merit function is created to assure global convergence of the new method. The modified LM method's cubic convergence is demonstrated under the local error bound condition, which is weaker than nonsingularity. The numerical results demonstrate that the new method is quite efficient and might save numerous Jacobian calculations.