On the Solution of Poissonxe2x80x99s Equation using Deep Learning We devise a numerical method for solving the Poissonxe2x80x99s equation using a convolutional neural network architecture, otherwise known as deep learning. The method we have employed here uses both feedforward neural systems and backpropagation to set up a framework for achieving the numerical solutions of the elliptic partial differential equations  more superficially the Poissonxe2x80x99s equation. Our deep learning framework has two substantial entities. The first part of the network enables to fulfill the necessary boundary conditions of the Poissonxe2x80x99s equation while the second part consisting of a feedforward neural system containing flexible parameters or weights gives rise to the solution. We have compared the solutions of the Poissonxe2x80x99s equation arising from our deep learning framework subject to various boundary conditions with the corresponding analytic solutions. As a result, we have found that our deep learning framework can obtain solutions which are accurate as well as efficient.

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.