  Developing accurate dynamical system models from physical insight or data can
be impeded when only partial observations of the system state are available.
Here, we combine conservation laws used in physics and engineering with
artificial neural networks to construct "grey-box" system models that make
accurate predictions even with limited information. These models use a time
delay embedding (c.f., Takens embedding theorem) to reconstruct effect of the
intrinsic states, and can be used for multiscale systems where macroscopic
balance equations depend on unmeasured micro/meso scale phenomena. By
incorporating physics knowledge into the neural network architecture, we
regularize variables and may train the model more accurately on smaller data
sets than black-box neural network models. We present numerical examples from
biotechnology, including a continuous bioreactor actuated using light through
optogenetics (an emerging technology in synthetic biology) where the effect of
unmeasured intracellular information is recovered from the histories of the
measured macroscopic variables.
