Neural Network Approximation for Nonlinear Partial Differential Equations with QuasiNewton Optimization and Piecewise Strategy In this paper, we propose a neural network based approach to approximate numerical solutions of partial differential equations (PDEs). Concretely, we understand the solution of PDEs as finding a reasonable mapping from the PDEs with initial and boundary conditions to the space of solutions. Then, we approximate this mapping by a neural network with best fitting the data from the PDEs, where the initial and boundary conditions are cooperated to the loss function. The main contributions of proposed method are twofold. First, we adopt the quasiNewton algorithm to minimize the loss instead of the stochastic gradient method, which is benefit from the relatively low sampling rate. It avoids the instability arising from the stochastic method. On the other hand, we adopt the piecewise strategy to improve the computational efficiency. To further understand the approximate process, we also investigate how the network architecture affect the performance. Finally, we compared the proposed method with some stateoftheart numerical methods for PDEs. The experimental results validate that the proposed method is more flexible and can avoid the difficulties caused by the complexity of target PDEs, and hence obtains the best performance, even for nonlinear equation with shock.

Novel design of Morlet wavelet neural network for solving second order LaneEmden equation Abstract   In this study, a novel computational paradigm based on Morlet wavelet neural network (MWNN) optimized with integrated strength of genetic algorithm (GAs) and Interiorpoint algorithm (IPA) is presented for solving second order LaneEmden equation (LEE). The solution of the LEE is performed by using modeling of the system with MWNNs aided with a hybrid combination of global search of GAs and an efficient local search of IPA. Three variants of the LEE have been numerically evaluated and their comparison with exact solutions demonstrate the correctness of the presented methodology. The statistical analyses are performed to establish the accuracy and convergence via the Theilxe2x80x99s inequality coefficient, mean absolute deviation, and Nash Sutcliffe efficiency based metrics.