论文标题
通过Relu DNNS的参数线性传输方程解决方案的有效近似
Efficient Approximation of Solutions of Parametric Linear Transport Equations by ReLU DNNs
论文作者
论文摘要
我们证明，具有RELU激活函数的深神经网络可以有效地近似于各种参数线性传输方程的解。对于非平滑的初始条件，这些PDE的溶液是高维和非平滑态的。因此，这些功能的近似受到维度的诅咒。我们证明，通过其固有的组成性，深神网络可以解决传输方程的特征流，从而使近似速率与参数维度无关。
We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation of these functions suffers from a curse of dimension. We demonstrate that through their inherent compositionality deep neural networks can resolve the characteristic flow underlying the transport equations and thereby allow approximation rates independent of the parameter dimension.
