Concrete crack opening forecasting by back propagation neural network and differential equation

Concrete crack opening (CCO) is of great importance to hydraulic engineering maintenance. A forecast method is put forward combining back propagation neural network (BPNN) and differential equation (DE) for daily CCO modeling and was applied to Wangqingtuo Reservoir, in the northern semiarid region of China and the contribution of the DE was assessed by using BPNN model as a contrast. First, it is made up of BPNN and DE calibrations: (1) use historical data to calibrate BPNN models and obtain residuals; (2) use the particle swarm optimization to calibrate coefficients of the DE. The periodicity and time delay of air temperature is expressed by the DE well. Second, important results were found by field application: (1) the sole BPNN models can provide reasonable predictions; (2) better prediction can be achieved based on BPNN-DE-2TD by increasing KGE, 12% for JB-1, 37% for JB-3, and 6% for JB-7; (3) it is indicated that the addition of DE can improve the modeling on the role of air temperature under seasonal and linear trend, while BPNN part can express the nonlinear role of water level and precipitation well, confirmed by Fourier amplitude sensitivity test sensitivity and Shapley Additive exPlanations analysis. This study could provide useful insights into further forecasting of CCO under this forecast method in the world.