This paper proposes the memristor-memristor (M-M) and the memristor-gradient descent (M-GD) neural networks based on the classical back propagation neural network. The presented models are employed to predict the compressive strength of high-performance hydraulic concrete (HPC), and are tested by well-fitting and accurate predictions with the experimental data. The developed algorithms are also evaluated through comparisons with the classical learning algorithms including the gradient descent method, the gradient descent with momentum, the gradient descent with adaptive learning rate, the elastic gradient descent, and the Levenberg-Marquardt algorithm. It is observed that the established M-GD generally outperforms the classical algorithms and M-M. The constructed M-M neural network has a quite high convergence speed, and the strength prediction error induced by it can roughly satisfy the demands in construction engineering. This work extends the nonlinear memristor to a brand-new field, and provides an effective methodology for forecasting the compressive strength prediction of HPC.