  Infrastructure monitoring is critical for safe operations and sustainability.
Water distribution networks (WDNs) are large-scale networked critical systems
with complex cascade dynamics which are difficult to predict. Ubiquitous
monitoring is expensive and a key challenge is to infer the contaminant
dynamics from partial sparse monitoring data. Existing approaches use
multi-objective optimisation to find the minimum set of essential monitoring
points, but lack performance guarantees and a theoretical framework.
  Here, we first develop Graph Fourier Transform (GFT) operators to compress
networked contamination spreading dynamics to identify the essential principle
data collection points with inference performance guarantees. We then build
autoencoder (AE) inspired neural networks (NN) to generalize the GFT sampling
process and under-sample further from the initial sampling set, allowing a very
small set of data points to largely reconstruct the contamination dynamics over
real and artificial WDNs. Various sources of the contamination are tested and
we obtain high accuracy reconstruction using around 5-10% of the sample set.
This general approach of compression and under-sampled recovery via neural
networks can be applied to a wide range of networked infrastructures to enable
digital twins.
