30 research outputs found
De Rham compatible Deep Neural Network FEM
On general regular simplicial partitions of bounded polytopal
domains , , 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
of are required for DNN emulation. In addition, for CPwL
functions our DNN construction is valid in any dimension . Our
``FE-Nets'' are required in the variationally correct, structure-preserving
approximation of boundary value problems of electromagnetism in nonconvex
polyhedra . 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
Convergent adaptive hybrid higher-order schemes for convex minimization
This paper proposes two convergent adaptive mesh-refining algorithms for the
hybrid high-order method in convex minimization problems with two-sided
p-growth. Examples include the p-Laplacian, an optimal design problem in
topology optimization, and the convexified double-well problem. The hybrid
high-order method utilizes a gradient reconstruction in the space of piecewise
Raviart-Thomas finite element functions without stabilization on triangulations
into simplices or in the space of piecewise polynomials with stabilization on
polytopal meshes. The main results imply the convergence of the energy and,
under further convexity properties, of the approximations of the primal resp.
dual variable. Numerical experiments illustrate an efficient approximation of
singular minimizers and improved convergence rates for higher polynomial
degrees. Computer simulations provide striking numerical evidence that an
adopted adaptive HHO algorithm can overcome the Lavrentiev gap phenomenon even
with empirical higher convergence rates