The present investigations are related to design an integrated computing numerical approach through Levenberg–Marquardt backpropagation (LMB) neural networks (NNs), i.e., LMB-NNs. The designed LMB-NNs approach is presented to solve the fourth-order nonlinear system of Emden–Fowler model (FO-SEFM). The solution of six different examples based on the FO-SEFM using the designed methodology LMB-NNs is numerically treated along with the discussion of singular point and shape factor. The comparison of the obtained results from the LMB-NNs and the exact solutions of each example has been presented. To evaluate the approximate results of the FO-SEFM for different problems, the testing, training, and authentication procedures are accompanied to adapt the NNs by reducing the functions of mean square error (MSE) through the LMB. The proportional investigations and performance studies based on the results of error histograms, MSE, regression, and correlation establish the effectiveness and correctness of the designed LMB-NNs approach.

目前的研究涉及通过 Levenberg-Marquardt 反向传播 (LMB) 神经网络 (NNs)，即 LMB-NNs 设计集成计算数值方法。提出了设计的 LMB-NNs 方法来解决 Emden-Fowler 模型 (FO-SEFM) 的四阶非线性系统。使用设计的 LMB-NNs 方法对基于 FO-SEFM 的六个不同示例的解决方案进行了数值处理，并讨论了奇异点和形状因子。已经给出了从 LMB-NN 获得的结果与每个示例的精确解的比较。为了评估不同问题的 FO-SEFM 的近似结果，测试、训练和认证过程伴随着通过 LMB 减少均方误差 (MSE) 函数来适应 NN。