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Simultaneous neural network approximation for smooth functions
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DOI：10.1016/j.neunet.2022.06.040.png)
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Abstract
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We establish in this work approximation results of deep neural networks for smooth functions measured in Sobolev norms, motivated by recent development of numerical solvers for partial differential equations using deep neural networks. Our approximation results are nonasymptotic in the sense that the error bounds are explicitly characterized in terms of both the width and depth of the networks simultaneously with all involved constants explicitly determined. Namely, for f is an element of C-s ([0, 1](d)), we show that deep ReLU networks of width O(N log N) and of depth O(L log L) can achieve a nonasymptotic approximation rate of O(N-2(s-1)/d L-2(s-1)/d) with respect to the W-1,W-p([0, 1](d)) norm for p is an element of [1, infinity). If either the ReLU function or its square is applied as activation functions to construct deep neural networks of width O(N log N) and of depth O(L log L) to approximate f is an element of C-s ([0, 1](d)), the approximation rate is O((N-2(s-n)/dL-2(s-n)/d) with respect to the W-n,W-p([0, 1](d)) norm for p is an element of [1, infinity). (C) 2022 Elsevier Ltd. All rights reserved.
Keywords:
Deep neural networks
Sobolev norm
ReLUk activation functions
Approximation theory
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