A semi discrete convolutional neural network algorithm for solving hyperbolic equations(求解双曲型方程的半离散卷积神经网络算法)

Abstract

The existence of discontinuous solutions of the hyperbolic conservation laws calls for require strict numerical algorithms. Traditional low order numerical algorithms are usually easy to construct, but the resolution of the numerical results is low and depends on the grid. Machine learning methods do not rely on grids and are suitable for dealing with complex scenarios (such as high-dimensional problems), but there may be displacement or smoothing phenomena when solving discontinuous problems. By combining machine learning with traditional low order formats and leveraging their advantages, this paper adopts a low order finite volume format, and uses convolutional neural networks to optimize weight coefficients. High resolution numerical results can be obtained by solving with fewer nodes. We demonstrate the good performance of the algorithm through several numerical examples which shows that the algorithm can obtain high-resolution numerical results for both continuous and discontinuous problems without any displacement or smoothing phenomena.(双曲守恒律方程的间断解对数值算法要求严格。通常传统低阶数值算法构造简单，但数值结果的分辨率较低，且依赖于网格。机器学习方法虽不依赖于网格，且适用于处理复杂场景下的问题（如高维问题），但在求间断解时可能出现移位或抹平现象。将机器学习方法与传统低阶格式相结合，在空间上采用低阶有限体积格式，在权重系数优化上采用卷积神经网络，从而实现基于较少节点便可得到高分辨率数值结果。数值算例验证了算法的性能，在求解连续及间断问题时均能得到高分辨率数值结果，且未出现移位、抹平现象。)

Keywords

• hyperbolic conservation law equation(双曲守恒律方程)
• finite volume method(有限体积法)
• convolutional neural network model(卷积神经网络模型)