  We are interested in the approximation of partial differential equations with
a data-driven approach based on the reduced basis method and machine learning.
We suppose that the phenomenon of interest can be modeled by a parametrized
partial differential equation, but that the value of the physical parameters is
unknown or difficult to be directly measured. Our method allows to estimate
fields of interest, for instance temperature of a sample of material or
velocity of a fluid, given data at a handful of points in the domain. We
propose to accomplish this task with a neural network embedding a reduced basis
solver as exotic activation function in the last layer. The reduced basis
solver accounts for the underlying physical phenomenonon and it is constructed
from snapshots obtained from randomly selected values of the physical
parameters during an expensive offline phase. The same full order solutions are
then employed for the training of the neural network. As a matter of fact, the
chosen architecture resembles an asymmetric autoencoder in which the decoder is
the reduced basis solver and as such it does not contain trainable parameters.
The resulting latent space of our autoencoder includes parameter-dependent
quantities feeding the reduced basis solver, which -- depending on the
considered partial differential equation -- are the values of the physical
parameters themselves or the affine decomposition coefficients of the
differential operators.
