  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.
