  This work introduces Physics-informed State-space neural network Models
(PSMs), a novel solution to achieving real-time optimization, flexibility, and
fault tolerance in autonomous systems, particularly in transport-dominated
systems such as chemical, biomedical, and power plants. Traditional data-driven
methods fall short due to a lack of physical constraints like mass
conservation; PSMs address this issue by training deep neural networks with
sensor data and physics-informing using components' Partial Differential
Equations (PDEs), resulting in a physics-constrained, end-to-end differentiable
forward dynamics model. Through two in silico experiments -- a heated channel
and a cooling system loop -- we demonstrate that PSMs offer a more accurate
approach than a purely data-driven model. In the former experiment, PSMs
demonstrated significantly lower average root-mean-square errors across test
datasets compared to a purely data-driven neural network, with reductions of 44
%, 48 %, and 94 % in predicting pressure, velocity, and temperature,
respectively.
  Beyond accuracy, PSMs demonstrate a compelling multitask capability, making
them highly versatile. In this work, we showcase two: supervisory control of a
nonlinear system through a sequentially updated state-space representation and
the proposal of a diagnostic algorithm using residuals from each of the PDEs.
The former demonstrates PSMs' ability to handle constant and time-dependent
constraints, while the latter illustrates their value in system diagnostics and
fault detection. We further posit that PSMs could serve as a foundation for
Digital Twins, constantly updated digital representations of physical systems.
