A Differentiable Distribution-Centric Gas Chromatographic Model for Gradient-Based Optimization of Temperature Programs

Abstract

Blumberg’s distribution-centric model, the analyte migration integral and a band-broadening layer based on the Golay equation are reformulated in a differentiable form and validated against a published simulation for the 16 EPA priority PAHs. The resolution of every peak pair is thereby an explicit differentiable function of the temperature program, and the exact gradient of any criterion built from these resolutions is returned by backpropagation through the simulator, irrespective of the number of program parameters. Separation conditions can therefore be selected purposefully, for programs of any dimensionality. The method returns the attainable front of the separation-time trade-off rather than a single program; the three forms of objective function compared here reach different regions of that front, so the criterion stays a free part of the formulation, written to suit the analysis at hand. Pairs that no program brings to baseline separation are identified from the search and excluded from the criterion. At an equal number of calls to the model, backpropagation reaches an objective value that direct search, an evolution strategy and finite differences do not attain. The attainable separation improves as the profile is allowed more nodes, and the optimizer does not reproduce the conventional shape of a few linear ramps, it smooths the heating where the separation physics rewards it. The number of ramps that current instruments accept is a restriction of their control software, and it limits the separation quality the same column could otherwise deliver.