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\'{e}d\'{e}lec 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.

翻译：在一般有界多面体区域 $\Omega \subset \mathbb{R}^d$（$d\in\{2,3\}$）的规范单纯形划分 $\mathcal{T}$ 上，我们构造了离散德·拉姆复形中所有最低阶有限元空间的精确神经网络模拟。这些空间包括分片常数函数空间、连续分片线性函数空间、经典的“Raviart-Thomas单元”和“Nédélec边单元”。除连续分片线性函数情况外，我们的网络架构同时采用ReLU（修正线性单元）和BiSU（二值阶跃单元）激活函数来捕捉不连续性。在连续分片线性函数这一重要情形中，我们证明了仅使用纯ReLU网络即可完成实现。与先前结果相比，我们的构造和深度神经网络架构具有更广泛的适用性——对 $\Omega$ 的规范单纯形划分 $\mathcal{T}$ 无需施加任何几何约束即可实现深度神经网络模拟。此外，对于连续分片线性函数，我们的深度神经网络构造在任意维度 $d\geq 2$ 均成立。在非凸多面体 $\Omega \subset \mathbb{R}^3$ 的电磁边值问题的变分正确、结构保持逼近中，所提出的“有限元网络”具有基础性作用。因此，它们是通过深度学习技术进行电磁场模拟时，应用“物理信息神经网络”或“深度Ritz方法”等框架的关键要素。本文还指出了本构造向高阶相容空间及其他非相容离散化类别（特别是“Crouzeix-Raviart”单元和混合高阶方法）的推广方向。