  Petrov-Galerkin formulations with optimal test functions allow for the
stabilization of finite element simulations. In particular, given a discrete
trial space, the optimal test space induces a numerical scheme delivering the
best approximation in terms of a problem-dependent energy norm. This ideal
approach has two shortcomings: first, we need to explicitly know the set of
optimal test functions; and second, the optimal test functions may have large
supports inducing expensive dense linear systems. Nevertheless, parametric
families of PDEs are an example where it is worth investing some (offline)
computational effort to obtain stabilized linear systems that can be solved
efficiently, for a given set of parameters, in an online stage. Therefore, as a
remedy for the first shortcoming, we explicitly compute (offline) a function
mapping any PDE-parameter, to the matrix of coefficients of optimal test
functions (in a basis expansion) associated with that PDE-parameter. Next, as a
remedy for the second shortcoming, we use the low-rank approximation to
hierarchically compress the (non-square) matrix of coefficients of optimal test
functions. In order to accelerate this process, we train a neural network to
learn a critical bottleneck of the compression algorithm (for a given set of
PDE-parameters). When solving online the resulting (compressed) Petrov-Galerkin
formulation, we employ a GMRES iterative solver with inexpensive matrix-vector
multiplications thanks to the low-rank features of the compressed matrix. We
perform experiments showing that the full online procedure as fast as the
original (unstable) Galerkin approach. In other words, we get the stabilization
with hierarchical matrices and neural networks practically for free. We
illustrate our findings by means of 2D Eriksson-Johnson and Hemholtz model
problems.
