De Rham compatible Deep Neural Network FEM.

Area of Science:

• Computational Electromagnetics
• Numerical Analysis
• Deep Learning

Background:

• Finite element methods (FEM) are crucial for solving complex electromagnetic field problems.
• Existing deep learning approaches for FEM often face limitations with geometric complexity and specific element types.

Purpose of the Study:

• To construct exact neural network (NN) emulations of finite element spaces.
• To generalize these emulations to arbitrary regular simplicial partitions and higher dimensions.
• To enable structure-preserving approximations for electromagnetic boundary value problems.

Main Methods:

• Development of novel neural network architectures, termed "FE-Nets".
• Utilizing ReLU (rectified linear unit) and BiSU (binary step unit) activations for discontinuous functions.
• Proving the sufficiency of pure ReLU nets for continuous piecewise linear (CPwL) functions.
• Demonstrating applicability to Raviart-Thomas and Nédélec elements.

Main Results:

• Exact NN emulations of lowest order finite element spaces are achieved.
• The construction is general, requiring no geometric restrictions on partitions.
• CPwL function emulation is valid in any dimension d≥2.
• FE-Nets are applicable to nonconvex polyhedra in 3D.

Conclusions:

• FE-Nets provide a foundation for physics-informed NNs and deep Ritz methods in electromagnetics.
• These NNs facilitate accurate, structure-preserving approximations of electromagnetic fields.
• The methodology offers a pathway for advanced deep learning applications in computational physics.