  We propose a method, based on Artificial Neural Networks, that learns the
dependence of the constant in the Poincar\'e inequality on polygonal elements
of Voronoi meshes, on some geometrical metrics of the element. The cost of this
kind of algorithms mainly resides in the data preprocessing and learning
phases, that can be performed offline once and for all, constructing an
efficient method for computing the constant, which is needed in the design of a
posteriori error estimates in numerical mesh-based schemes for the solution of
Partial Differential Equations.
