On the approximation of rough functions with deep neural networks

Understand

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.