Fluid flow and heat transfer of a second-order viscoelastic fluid in an axisymmetric channel with a porous wall for turbine cooling applications are studied. The nonlinear differential equations of the fluid flow and heat transfer arising from similarity solutions are computed employing a Hybrid Neural Network-Particle Swarm Optimization algorithm (HNNPSO). A trial function, satisfying the boundary conditions, as a possible solution for the governing equations is introduced. The trial functions incorporate a multi-layer perceptron neural network with adjustable parameters (the weights and biases). The Particle Swarm Optimization algorithm (PSO) is applied to find the adjustable parameters of the trial solution to satisfy the governing equations. Finally, comparisons are made between the results of the present method (HNNPSO) and the results of the fourth order Runge–Kutta method, finite difference method, and Variational Iteration Method. The results indicate that HNNPSO method conveniently produces a polynomial analytic solution with remarkable accuracy, and the accuracy of the solution improves as the number of neurons of the neural network increases.