Wavelet neural networks based solutions for elliptic partial differential equations with improved butterfly optimization algorithm training

Abstract In this study, a machine learning approach based on the unsupervised version of wavelet neural networks (WNNs) is used to solve two-dimensional elliptic partial differential equations (PDEs). The design of the WNNs must be judiciously addressed, particularly, the adopted training algorithm, since it greatly influences the generalization performance and the convergence rate of WNNs. Although the gradient information of the commonly used gradient descent training algorithm in WNNs may direct the search to optimal weight solutions that minimize the error function, the learning process is slow due to the complex calculation of the partial derivatives. To date, on account of the derivative free characteristic and adaptability to respond to the complex dynamic changes of the interdependencies, numerous studies explored the potential benefit of integrating a meta-heuristic algorithm as the training algorithm of WNNs, where encouraging results are achieved. In this paper, an improved butterfly optimization algorithm (IBOA) is proposed and subsequently integrated into the training process of the WNNs. To evaluate the performance of the proposed IBOA training method, the obtained results are compared to the results of the momentum backpropagation (MBP), the particle swarm optimization (PSO) and the standard butterfly optimization algorithm (BOA) training methods. Statistical analyses of the results based on a sufficient number of independent runs validate the effectiveness of the proposed method in terms of accuracy, robustness and convergence.