Exponential Expressivity of ReLU$^k$ Neural Networks on Gevrey Classes with Point Singularities
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arXiv
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| Format: | Preprint |
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2024
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| author | Opschoor, Joost A. A. Schwab, Christoph |
| author_facet | Opschoor, Joost A. A. Schwab, Christoph |
| contents | We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains $\mathrm{D} \subset \mathbb{R}^d$, $d=2,3$. We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in $\mathrm{D}$, comprising the countably-normed spaces of I.M. Babuška and B.Q. Guo.
As intermediate result, we prove that continuous, piecewise polynomial high order (``$p$-version'') finite elements with elementwise polynomial degree $p\in\mathbb{N}$ on arbitrary, regular, simplicial partitions of polyhedral domains $\mathrm{D} \subset \mathbb{R}^d$, $d\geq 2$ can be exactly emulated by neural networks combining ReLU and ReLU$^2$ activations. On shape-regular, simplicial partitions of polytopal domains $\mathrm{D}$, both the number of neurons and the number of nonzero parameters are proportional to the number of degrees of freedom of the finite element space, in particular for the $hp$-Finite Element Method of I.M. Babuška and B.Q. Guo. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_02035 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exponential Expressivity of ReLU$^k$ Neural Networks on Gevrey Classes with Point Singularities Opschoor, Joost A. A. Schwab, Christoph Numerical Analysis Machine Learning 65N30, 41A25 We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains $\mathrm{D} \subset \mathbb{R}^d$, $d=2,3$. We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in $\mathrm{D}$, comprising the countably-normed spaces of I.M. Babuška and B.Q. Guo. As intermediate result, we prove that continuous, piecewise polynomial high order (``$p$-version'') finite elements with elementwise polynomial degree $p\in\mathbb{N}$ on arbitrary, regular, simplicial partitions of polyhedral domains $\mathrm{D} \subset \mathbb{R}^d$, $d\geq 2$ can be exactly emulated by neural networks combining ReLU and ReLU$^2$ activations. On shape-regular, simplicial partitions of polytopal domains $\mathrm{D}$, both the number of neurons and the number of nonzero parameters are proportional to the number of degrees of freedom of the finite element space, in particular for the $hp$-Finite Element Method of I.M. Babuška and B.Q. Guo. |
| title | Exponential Expressivity of ReLU$^k$ Neural Networks on Gevrey Classes with Point Singularities |
| topic | Numerical Analysis Machine Learning 65N30, 41A25 |
| url | https://arxiv.org/abs/2403.02035 |