2019 IEEE 31st International Conference on Tools with Artificial Intelligence (ICTAI) | 2019
Neural Network Approximation for Nonlinear Partial Differential Equations with Quasi-Newton Optimization and Piecewise Strategy
Abstract
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 quasi-Newton 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 state-of-the-art 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.