论文标题
基于深度学习的非线性升级方法的传输方程
A deep learning based nonlinear upscaling method for transport equations
论文作者
论文摘要
我们将为非线性传输方程开发一种非线性升级方法。提出的方案给出了溶液平均值的粗略方程。为了计算粗尺度方程中的参数，构建了局部降尺度运算符。这种降尺度操作使用平均值恢复了精细的比例属性。这是通过以给定单元平均值作为约束的过采样区域上的方程来实现的。由于非线性，需要即时计算这些降尺度操作，并且不能预先计算这些数量。为了进行有效的降尺度操作，我们采用了深度学习方法。我们将使用深层神经网络来近似缩小操作。我们的数值结果表明，所提出的方案可以实现良好的准确性和效率。
We will develop a nonlinear upscaling method for nonlinear transport equation. The proposed scheme gives a coarse scale equation for the cell average of the solution. In order to compute the parameters in the coarse scale equation, a local downscaling operator is constructed. This downscaling operation recovers fine scale properties using cell averages. This is achieved by solving the equation on an oversampling region with the given cell average as constraint. Due to the nonlinearity, one needs to compute these downscaling operations on the fly and cannot pre-compute these quantities. In order to give an efficient downscaling operation, we apply a deep learning approach. We will use a deep neural network to approximate the downscaling operation. Our numerical results show that the proposed scheme can achieve a good accuracy and efficiency.
