Solving the Poisson Equation using Virtual Element Method and Artificial Neural Networks

Resumo

The Virtual Element Method (VEM) is a recent method that proposes a generalization of the classical
Finite Element Method (FEM). VEM formulation is complex and requires a strong mathematical base. VEM
models are able to use any simple polygon (convex or non-convex polygons) as mesh discretization elements,
which leads to the functions associated with each of those elements being not strictly polynomials. Thus, the
core proposal of VEM is to compute those functions implicitly, by projecting then in a polynomial space. The
Virtual Element Method was originally formulated for the Poisson Equation, which is largely explored both in
mathematics and natural sciences due to its versatility.
Approximation methods like VEM are widely applied to solve partial differential equations (PDE). A less
common approach is to use artificial intelligence techniques like Artificial Neural Networks (ANNs). Solve PDEs
with ANNs is not new, but with the increasing popularity of deep learning techniques and new frameworks (e.g

Pytorch and Keras), this approach is becoming more relevant. In the present work, the Poisson Equation is trans-
formed into an optimization problem and them ANNs are used to compose a trial solution, aiming to obtain the

best one (i.e., the one that reduces the error as much as possible). Artificial Neural Networks are widely used in
this kind of approach since they are very popular in classification problems.
Throughout the article, the formulation of VEM and the ANNs approach is presented, highlighting their main
characteristics. The implementation of both approaches is discussed and the results are compared.