  Thanks to their universal approximation properties and new efficient training
strategies, Deep Neural Networks are becoming a valuable tool for the
approximation of mathematical operators. In the present work, we introduce
Mesh-Informed Neural Networks (MINNs), a class of architectures specifically
tailored to handle mesh based functional data, and thus of particular interest
for reduced order modeling of parametrized Partial Differential Equations
(PDEs). The driving idea behind MINNs is to embed hidden layers into discrete
functional spaces of increasing complexity, obtained through a sequence of
meshes defined over the underlying spatial domain. The approach leads to a
natural pruning strategy which enables the design of sparse architectures that
are able to learn general nonlinear operators. We assess this strategy through
an extensive set of numerical experiments, ranging from nonlocal operators to
nonlinear diffusion PDEs, where MINNs are compared against more traditional
architectures, such as classical fully connected Deep Neural Networks, but also
more recent ones, such as DeepONets and Fourier Neural Operators. Our results
show that MINNs can handle functional data defined on general domains of any
shape, while ensuring reduced training times, lower computational costs, and
better generalization capabilities, thus making MINNs very well-suited for
demanding applications such as Reduced Order Modeling and Uncertainty
Quantification for PDEs.
