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.