On the approximation of rough functions with deep neural networks
This work provides a novel connection between ENO procedures and neural networks, potentially enhancing approximation methods for rough functions in computational mathematics and data science.
The authors proved that the ENO interpolation procedure can be represented as a deep ReLU neural network at any order, enabling transfer of high-order accuracy for approximating Lipschitz functions. Numerical tests demonstrated excellent performance in approximating solutions of nonlinear conservation laws and data compression.
Deep neural networks and the ENO procedure are both efficient frameworks for approximating rough functions. We prove that at any order, the ENO interpolation procedure can be cast as a deep ReLU neural network. This surprising fact enables the transfer of several desirable properties of the ENO procedure to deep neural networks, including its high-order accuracy at approximating Lipschitz functions. Numerical tests for the resulting neural networks show excellent performance for approximating solutions of nonlinear conservation laws and at data compression.