  Many applications in computational physics involve approximating problems
with microstructure, characterized by multiple spatial scales in their data.
However, these numerical solutions are often computationally expensive due to
the need to capture fine details at small scales. As a result, simulating such
phenomena becomes unaffordable for many-query applications, such as
parametrized systems with multiple scale-dependent features. Traditional
projection-based reduced order models (ROMs) fail to resolve these issues, even
for second-order elliptic PDEs commonly found in engineering applications. To
address this, we propose an alternative nonintrusive strategy to build a ROM,
that combines classical proper orthogonal decomposition (POD) with a suitable
neural network (NN) model to account for the small scales. Specifically, we
employ sparse mesh-informed neural networks (MINNs), which handle both spatial
dependencies in the solutions and model parameters simultaneously. We evaluate
the performance of this strategy on benchmark problems and then apply it to
approximate a real-life problem involving the impact of microcirculation in
transport phenomena through the tissue microenvironment.
