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.

中文翻译：

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利用ReLU DNN高效逼近参数线性传输方程的解

我们证明了具有ReLU激活功能的深度神经网络可以有效地近似各种参数线性运输方程的解。对于不平滑的初始条件，这些PDE的解是高维且不平滑的。因此，这些函数的近似遭受尺寸的诅咒。我们证明了深层神经网络通过其固有的组成性可以解析运输方程背后的特征流，从而允许近似速率与参数维数无关。