  Comparison of echelle spectra to synthetic models has become a computational
statistics challenge, with over ten thousand individual spectral lines
affecting a typical cool star echelle spectrum. Telluric artifacts, imperfect
line lists, inexact continuum placement, and inflexible models frustrate the
scientific promise of these information-rich datasets. Here we debut an
interpretable machine-learning framework "blas\'e" that addresses these and
other challenges. The semi-empirical approach can be viewed as "transfer
learning" -- first pre-training models on noise-free precomputed synthetic
spectral models, then learning the corrections to line depths and widths from
whole-spectrum fitting to an observed spectrum. The auto-differentiable model
employs back-propagation, the fundamental algorithm empowering modern Deep
Learning and Neural Networks. Here, however, the 40,000+ parameters symbolize
physically interpretable line profile properties such as amplitude, width,
location, and shape, plus radial velocity and rotational broadening. This
hybrid data-/model- driven framework allows joint modeling of stellar and
telluric lines simultaneously, a potentially transformative step forwards for
mitigating the deleterious telluric contamination in the near-infrared. The
blas\'e approach acts as both a deconvolution tool and semi-empirical model.
The general purpose scaffolding may be extensible to many scientific
applications, including precision radial velocities, Doppler imaging, chemical
abundances, and remote sensing. Its sparse-matrix architecture and
GPU-acceleration make blas\'e fast. The open-source PyTorch-based code includes
tutorials, Application Programming Interface (API) documentation, and more. We
show how the tool fits into the existing Python spectroscopy ecosystem,
demonstrate a range of astrophysical applications, and discuss limitations and
future extensions.
