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De Rham compatible Deep Neural Network FEM

by M. Longo and J. A. A. Opschoor and N. Disch and Ch. Schwab and J. Zech

(Report number 2022-03)

Abstract
On general regular simplicial partitions \(\mathcal{T}\) of bounded polytopal domains \(\Omega \subset \mathbb{R}^d\), \(d\in\{2,3\}\), we construct \emph{exact neural network (NN) emulations} of all lowest order finite element spaces in the discrete de Rham complex. These include the spaces of piecewise constant functions, continuous piecewise linear (CPwL) functions, the classical ``Raviart-Thomas element'', and the ``N\'ed\'elec edge element''. For all but the CPwL case, our network architectures employ both ReLU (rectified linear unit) and BiSU (binary step unit) activations to capture discontinuities. In the important case of CPwL functions, we prove that it suffices to work with pure ReLU nets. Our construction and DNN architecture generalizes previous results in that no geometric restrictions on the regular simplicial partitions \(\mathcal{T}\) of \(\Omega\) are required for DNN emulation. In addition, for CPwL functions our DNN construction is valid in any dimension \(d\geq 2\). Our ``FE-Nets'' are required in the variationally correct, structure-preserving approximation of boundary value problems of electromagnetism in nonconvex polyhedra \(\Omega \subset \mathbb{R}^3\). They are thus an essential ingredient in the application of e.g., the methodology of ``physics-informed NNs'' or ``deep Ritz methods'' to electromagnetic field simulation via deep learning techniques. We indicate generalizations of our constructions to higher-order compatible spaces and other, non-compatible classes of discretizations, in particular the ``Crouzeix-Raviart'' elements and Hybridized, Higher Order (HHO) methods.

Keywords: De Rham Complex, Finite Elements, Neural Networks

@Techreport{LODSZ22_991,
  author = {M. Longo and J. A. A. Opschoor and N. Disch and Ch. Schwab and J. Zech},
  title = {De Rham compatible Deep Neural Network FEM},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-03},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-03.pdf },
  year = {2022}
}

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