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Abstract We present a general class of machine learning algorithms called parametric matrix models. In contrast with most existing machine learning models that imitate the biology of neurons, parametric matrix models use matrix equations that emulate physical systems. Similar to how physics problems are usually solved, parametric matrix models learn the governing equations that lead to the desired outputs. Parametric matrix models can be efficiently trained from empirical data, and the equations may use algebraic, differential, or integral relations. While originally designed for scientific computing, we prove that parametric matrix models are universal function approximators that can be applied to general machine learning problems. After introducing the underlying theory, we apply parametric matrix models to a series of different challenges that show their performance for a wide range of problems. For all the challenges tested here, parametric matrix models produce accurate results within an efficient and interpretable computational framework that allows for input feature extrapolation. 

A novel laser absorption sensing strategy has been developed to evaluate combustion progress through quantitative measurements of carbon dioxide (CO2) in high-pressure (> 50 atm), high-temperature (> 3000 K) hydrocarbon-fueled rocket combustion flows. The sensor enables a broad range of operability by probing rovibrational transitions in the bandhead of CO2 near 4.2 m, accessed with an interband cascade laser. Under extreme rocket conditions, this targeted bandhead region experiences line-mixing effects that favorably distort the molecular spectra. A preliminary spectroscopic model of line-mixing effects has been developed utilizing a high-enthalpy shock tube to achieve scalability of spectral simulations over a range of high temperatures and high pressures. The model is employed for quantitative interpretation of measured absorption signals. The mid-infrared light source was fiber-coupled for remote light delivery at propulsion test facilities. A wavelength modulation spectroscopy technique utilizing normalized-second harmonic detection was implemented for acquiring differential absorption signals in a harsh rocket combustor environment. Using this method, measurements of CO2 concentration have been demonstrated over a range of operating conditions up to 83 bar in a single-element-injector RP-2/GOx rocket combustor at the Air Force Research Laboratory in Edwards, CA.