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**Deep ReLU networks and high-order finite element methods.**
*(English)*
Zbl 1452.65354

Summary: Approximation rate bounds for emulations of real-valued functions on intervals by deep neural networks (DNNs) are established. The approximation results are given for DNNs based on ReLU activation functions. The approximation error is measured with respect to Sobolev norms. It is shown that ReLU DNNs allow for essentially the same approximation rates as nonlinear, variable-order, free-knot (or so-called “\(hp\)-adaptive”) spline approximations and spectral approximations, for a wide range of Sobolev and Besov spaces. In particular, exponential convergence rates in terms of the DNN size for univariate, piecewise Gevrey functions with point singularities are established. Combined with recent results on ReLU DNN approximation of rational, oscillatory, and high-dimensional functions, this corroborates that continuous, piecewise affine ReLU DNNs afford algebraic and exponential convergence rate bounds which are comparable to “best in class” schemes for several important function classes of high and infinite smoothness. Using composition of DNNs, we also prove that radial-like functions obtained as compositions of the above with the Euclidean norm and, possibly, anisotropic affine changes of co-ordinates can be emulated at exponential rate in terms of the DNN size and depth without the curse of dimensionality.

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65D07 | Numerical computation using splines |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

41A25 | Rate of convergence, degree of approximation |

41A46 | Approximation by arbitrary nonlinear expressions; widths and entropy |

35B65 | Smoothness and regularity of solutions to PDEs |

35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |

68T07 | Artificial neural networks and deep learning |

92B20 | Neural networks for/in biological studies, artificial life and related topics |

### Keywords:

deep neural networks; finite element methods; exponential covergence; Gevrey regularity; singularities
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XMLCite

\textit{J. A. A. Opschoor} et al., Anal. Appl., Singap. 18, No. 5, 715--770 (2020; Zbl 1452.65354)

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