In this study, a novel neuro-swarming computing solver is developed for numerical treatment of third-order nonlinear multi-singular Emden–Fowler equation (TONMS-EFE) by using function approximation ability of artificial neural networks (ANNs) modeling and global optimization mechanism of particle swarm optimization (PSO) integrated with local search of interior-point scheme (IPS), i.e., ANN-PSO-IPS. The inspiration for the design of ANN-PSO-IPS-based numerical solver comes with an objective of presenting a reliable, accurate and viable structure that combines the strength of ANNs optimized with the integrated soft computing frameworks to deal with such challenging systems based on TONMS-EFE. The proposed ANN-PSO-IPS is implemented for four variants of TONMS-EFEs, and comparison with exact solutions relieved its robustness, correctness and effectiveness, which is further authenticated through statistical explorations.