On the Approximation of Rough Functions with Artificial Neural Networks
Tim De Ryck · Repository for Publications and Research Data (ETH Zurich) · 2020
Deep neural networks and the ENO procedure are both efficient frameworks for approximating rough functions.We prove that at any order, the stencil shifts of the ENO and ENO-SR interpolation procedures can be exactly obtained using a deep ReLU neural network.In addition, we construct and provide error bounds for ReLU neural networks that directly approximate the output of the ENO and ENO-SR interpolation procedures.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 interpolating rough functions, data compression and approximating solutions of nonlinear conservation laws.i