Machine-learning-assisted retrieval of four temperatures from CO₂ nonequilibrium transmittance spectra

Accurate retrieval of rotational and vibrational temperatures from non-equilibrium CO₂ spectra is essential for diagnosing discharge plasmas, monitoring combustion flows, and modeling Mars entry aerothermodynamics. However, retrieving multiple temperature parameters from CO₂ transmittance spectra constitutes a strongly nonlinear and ill-posed inverse problem. Conventional iterative spectral fitting methods are computationally expensive, sensitive to initial guesses, and prone to local-minimum trapping, making stable four-temperature retrieval challenging. To address these limitations, a non-iterative surrogate inverse framework based on a multilayer perceptron (MLP) is proposed for direct mapping from CO₂ transmittance spectra to four-temperature states. A large-scale database containing 810,000 spectrum-temperature pairs was constructed using a line-by-line forward model based on RADIS and the Treanor non-equilibrium distribution. The input consists of 401-dimensional transmittance spectra in the 2284.2-2284.6 cm^-1 range, while the outputs are the rotational temperature T_rot and three vibrational temperatures T_v1, T_v2, and T_v3. Under noise-free conditions, the proposed model achieved mean absolute errors of 4.1 K, 7.5 K, 5.1 K, and 13.6 K, respectively, with all R² values exceeding 0.998. Even under 3% Gaussian noise, the R² values remained above 0.996. As a proof-of-concept demonstration, experimental validation on a CO₂ discharge spectrum shows that the MLP retrievals agree with RADIS literature values (RMSE = 0.0387). Notably, the MLP method reduces the average retrieval time from 68 s to 0.012 s per spectrum, achieving a speedup of approximately 5700 times. This work is presented as an application study that addresses a specific diagnostic bottleneck - real-time four-temperature retrieval.