The Artificial Neural Network (ANN) has emerged as an essential element in the modern structural optimization due to its outstanding ability to approximate nonlinear structural behavior and drastically reduce computation cost involved in numerous finite element analyses. The traditional optimization algorithm for truss structures usually involves thousands of structural analyses in the process of searching solutions, which implies substantial computation cost and limited application to large-size engineering optimization tasks. Recent research results in machine learning, surrogate modeling, and data-driven optimization show the possibility of replacing numerically expensive structural analyses with neural network-based model without loss of the solution accuracy. The work studies the utilization of Backpropagation Neural Network (BPNN) and Counterpropagation Neural Network (CPNN) models as surrogate structural analyzers for the weight optimization of truss structures. The two types of neural networks are trained on the dataset of structural responses obtained in the process of finite element analysis and incorporated into the optimization process. The performance of BPNN and CPNN models is analyzed according to computation efficiency, convergence properties, and approximation accuracy on a benchmark 52-member space truss. Moreover, the review of modern developments in machine learning-assisted structural optimization is presented. Both of the neural network architectures have shown considerable improvements in computation times while sustaining adequate accuracy of predictions. While the CPNN shows greater speed of convergence, BPNN shows better accuracy of structural responses predictions during the optimization process. This approach proves that application of ANN models for surrogate modeling can be considered as an effective and robust method of solving challenging structural optimization problems.