  Finding extended hydrodynamics equations valid from the dense gas region to
the rarefied gas region remains a great challenge. The key to success is to
obtain accurate constitutive relations for stress and heat flux. Data-driven
models offer a new phenomenological approach to learning constitutive relations
from data. Such models enable complex constitutive relations that extend
Newton's law of viscosity and Fourier's law of heat conduction by regression on
higher derivatives. However, the choices of derivatives in these models are
ad-hoc without a clear physical explanation. We investigated data-driven models
theoretically on a linear system. We argue that these models are equivalent to
non-linear length scale scaling laws of transport coefficients. The equivalence
to scaling laws justified the physical plausibility and revealed the limitation
of data-driven models. Our argument also points out that modeling the scaling
law could avoid practical difficulties in data-driven models like derivative
estimation and variable selection on noisy data. We further proposed a
constitutive relation model based on scaling law and tested it on the
calculation of Rayleigh scattering spectra. The result shows our data-driven
model has a clear advantage over the Chapman-Enskog expansion and moment
methods.
