Exact Representation of Piecewise Polynomial Finite Element Functions by Two-Hidden-Layer Multilayer Perceptrons
Abstract
We construct exact representations of finite element functions of arbitrary polynomial degree on simplicial meshes in any spatial dimension by two-hidden-layer multilayer perceptrons (MLPs). The first hidden layer uses ReLU0+ReLU, and the second uses the degree-dependent activation ReLUk. The neural networks can realize both zero-skeleton and prescribed-skeleton representatives of discontinuous finite element functions, where the former vanish on the mesh skeleton and the latter admit independently prescribed polynomial data on each relatively open lower-dimensional mesh subsimplex; in particular, continuous finite element functions can be represented pointwise on the closed domain. In the continuous case, a half-open decomposition reduces the second-hidden-layer width by assigning each subsimplex to a single incident element. The number of nonzero parameters grows linearly with the number of elements for fixed spatial dimension, polynomial degree, and output dimension, and it is of the same order as the number of global degrees of freedom of the finite element space in the continuous case, provided that the mesh is shape-regular. All parameters in the neural network can be explicitly computed via the information of the mesh and finite element function without training.
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BibTeX
@misc{hu2026exact,
title = {{Exact Representation of Piecewise Polynomial Finite Element Functions by Two-Hidden-Layer Multilayer Perceptrons}},
author = {Jun Hu and Pengzhan Jin and Shengyang Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39856},
eprint = {2609.39856},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39856},
}
RIS
TY - UNPB
TI - Exact Representation of Piecewise Polynomial Finite Element Functions by Two-Hidden-Layer Multilayer Perceptrons
AU - Jun Hu
AU - Pengzhan Jin
AU - Shengyang Xu
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39856
UR - https://arxiv.org/abs/2609.39856
AB - We construct exact representations of finite element functions of arbitrary polynomial degree on simplicial meshes in any spatial dimension by two-hidden-layer multilayer perceptrons (MLPs). The first hidden layer uses ReLU^0+ReLU, and the second uses the degree-dependent activation ReLU^k. The neural networks can realize both zero-skeleton and prescribed-skeleton representatives of discontinuous finite element functions, where the former vanish on the mesh skeleton and the latter admit independently prescribed polynomial data on each relatively open lower-dimensional mesh subsimplex; in particular, continuous finite element functions can be represented pointwise on the closed domain. In the continuous case, a half-open decomposition reduces the second-hidden-layer width by assigning each subsimplex to a single incident element. The number of nonzero parameters grows linearly with the number of elements for fixed spatial dimension, polynomial degree, and output dimension, and it is of the same order as the number of global degrees of freedom of the finite element space in the continuous case, provided that the mesh is shape-regular. All parameters in the neural network can be explicitly computed via the information of the mesh and finite element function without training.
ER -