Comparison of échelle spectra to synthetic models has become a computational statistics challenge, with over ten thousand individual spectral lines affecting a typical cool star échelle spectrum. Telluric artifacts, imperfect line lists, inexact continuum placement, and inflexible models frustrate the scientific promise of these information-rich datasets. Here we debut a scalable machine-learning framework "blasé" that addresses these 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 noisy whole-spectrum fitting. The auto-differentiable model employs back-propagation, the fundamental algorithm empowering modern Neural Networks. Here, however, the 40,000+ parameters symbolize physically interpretable line profile properties (amplitude, width, location, shape) plus RV and vsini, rather than difficult-to-interpret neural network "weights". This hybrid data-/model- driven framework allows joint modeling of stellar and telluric lines simultaneously, a potentially transformative step forwards for pesky telluric mitigation in the near-infrared. Blasé also acts as a deconvolution tool. It is suitable for Doppler Imaging scenarios with longitudinally symmetric surface features like bands or polar spots, which evade detection in differential techniques. Blasé can also create super-resolution semi-empirical templates useful for critically evaluating atomic and molecular line lists. Its sparse-matrix architecture and GPU-acceleration make blasé fast, with end-to-end training of 50+ échelle orders in under 1 minute. The open-source PyTorch-based code (blase.readthedocs.io) includes tutorials and API documentation. We show how the tool fits into the existing Python spectroscopy ecosystem, demonstrate a range of astrophysical applications, and discuss limitations and future extensions.
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