# De Rham compatible Deep Neural Network FEM

arXiv: https://arxiv.org/abs/2201.05395 · HTML: https://mlsanity.com/paper/2201.05395/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 40 references that are in the map.
- First posted on arXiv 14 Jan 2022.

## Builds on (most cited first, 8 of 8)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Why Deep Neural Networks for Function Approximation?](https://mlsanity.com/paper/1610.04161/) (2016)
- [Deep vs. shallow networks : An approximation theory perspective](https://mlsanity.com/paper/1608.03287/) (2016)
- [PowerNet: Efficient Representations of Polynomials and Smooth Functions by Deep Neural Networks with Rectified Power Units](https://mlsanity.com/paper/1909.05136/) (2019)

## Built on by (most cited first, 4 of 4)

- [Deep Operator Network Approximation Rates for Lipschitz Operators](https://mlsanity.com/paper/2307.09835/) (2023)
- [Neural and spectral operator surrogates: unified construction and expression rate bounds](https://mlsanity.com/paper/2207.04950/) (2022)
- [On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theorem](https://mlsanity.com/paper/2308.05509/) (2023)
- [Deep ReLU networks and high-order finite element methods II: Chebyshev emulation](https://mlsanity.com/paper/2310.07261/) (2023)

## Abstract (arXiv)

On general regular simplicial partitions T of bounded polytopal domains Omega subset ℝd, d∈2,3, we construct 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 T of Omega are required for DNN emulation. In addition, for CPwL functions our DNN construction is valid in any dimension d≥ 2. Our ``FE-Nets'' are required in the variationally correct, structure-preserving approximation of boundary value problems of electromagnetism in nonconvex polyhedra Omega subset ℝ3. 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.