Complexity-Based Structural Optimization of Deep Belief Network and Application in Wastewater Treatment Process

Deep belief network (DBN) is an effective deep learning model, which can learn the complex data by extracting features hierarchically. However, the successful application of DBN depends on the suitable size of the structure (the number of hidden neurons), which is still an open problem. Currently, the network structure size is basically determined by experience with a time-consuming process. In this article, a complexity-based structural optimization (CBSO) algorithm, based on multiobjective ordinal optimization (MOO), is developed for designing the DBN structure. First, the problem formulation of structural optimization of DBN is given, where the multiple objectives are to minimize the fitting error and complexity. Second, the lower bound for alignment probability in optimizing DBN structure is developed according to MOO. Finally, an effective method to maximize the probability of correct select is given to pursue the good tradeoff between the complexity and the performance. The performance of proposed CBSO algorithm is demonstrated via predicting and controlling water quality of wastewater treatment process (WWTP) using the CBSO-DBN-based model predictive control (MPC) strategy. The simulation results show that the resulting CBSO-DBN can find the better structure design by using CBSO algorithm with smaller fitting error and limited computational complexity, and thereby achieve the better performance in WWTP than its peers. Especially, the CBSO-DBN-MPC improves the control accuracy by 76.16% and computational complexity by 50.45%, respectively.