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We demonstrate that deep neural networks with the ReLU activation functioncan efficiently approximate the solutions of various types of parametric lineartransport equations. For non-smooth initial conditions, the solutions of thesePDEs are high-dimensional and non-smooth. Therefore, approximation of thesefunctions suffers from a curse of dimension. We demonstrate that through theirinherent compositionality deep neural networks can resolve the characteristicflow underlying the transport equations and thereby allow approximation ratesindependent of the parameter dimension.

Efficient Approximation of Solutions of Parametric Linear Transport Equations by ReLU DNNs