Mid-infrared hyperspectral emission measurements provide wide-band, highly detailed spectral information, enabling spatial distribution retrieval of multiple scalar values in combustion flames. However, inferring temperature and species concentrations from these spectra poses significant challenges due to the nonlinear, ill-posed, and potentially high-dimensional nature of the related inverse problems. The ill-posedness can result in slow convergence and high sensitivity to initial parameter guesses and experimental noise. It is often necessary to introduce appropriate prior information and apply regularization constraints to yield physically reasonable and stable retrieval results. However, in the reconstruction of multiple fields, it is challenging to accurately define prior conditions and effectively incorporate them into the model. Additionally, the choice of regularization methods and the tuning of parameters significantly influence the retrieval results. As a result, traditional methods struggle to achieve accurate and simultaneous reconstruction of temperature and multi-species concentrations in combustion fields. Artificial neural networks provide a promising solution by learning complex, implicit relationships between input and output data without explicit modeling of the underlying physical and chemical laws required. This capability makes them particularly well-suited for nonlinear inverse problems. However, while data-driven approaches have shown potential in solving various inverse problems, they often rely on extensive datasets from experiments or simulations to build a supervised training database. Consequently, neural networks are frequently treated as “black boxes”, with their predictive capability heavily dependent on the training data. While these models may generalize well within data-rich regions, they often struggle with accurate predictions for data outside the training distribution. Meanwhile, transferring a neural network trained on simulation datasets to predict experimental data faces challenges such as insufficient generalization ability and a lack of physical interpretability. To this end, it is essential to develop a robust framework that combines the strengths of artificial neural networks with the fundamental physical constraints of the problem.

We present a physics-based neural network model for inverse radiation, which is designed to retrieve temperature and multi-species mole fractions within a flame by combining mid-infrared emission spectroscopy measurements. The model integrates the physical information of the measurement system into the neural network’s training process, ensuring that the optimization objectives not only match the measurement data but also comply with the physical equations. This approach eliminates the need for training datasets or complex retrieval algorithms. Firstly, the model is adopted to retrieve the temperature, CO₂, CO, and H₂O mole fractions, as well as soot volume fraction distributions in a simulated ethylene laminar diffusion flame. To validate the model’s accuracy and robustness, we add 10% Gaussian random noise to the simulated emission spectra. Furthermore, to experimentally validate the retrieval model, we conduct an ethylene laminar diffusion flame combustion experiment and measure the mid-infrared emission spectra of the experimental flame by utilizing a Fourier transform infrared spectrometer (FTIR). The emission spectra are calibrated by employing a blackbody furnace. Based on the proposed physics-based neural network model, the multiple scalar field distributions of the experimental flame are reconstructed.

The reconstruction results of the simulated flame show that by employing six radial projections, the physics-based neural network achieves sound agreement with the reference solution even with 10% random noise (Figs. 7 and 8). This demonstrates that spectral measurements provide valuable information for resolving the spatial distribution of scalar fields and the proposed model exhibits good robustness in handling noisy spectral data. The retrieval accuracy for temperature is higher than that for gases and soot, as the radiative properties of all three gases are nonlinearly related to temperature. Additionally, temperature governs blackbody radiation, amplifying its effect on the spectral signal. The model can also retrieve soot distribution with limited spatial projections. As soot and gas radiations are coupled in gas spectral regions, the distinct spectral features of gases lead to significant changes in the mixture’s radiations with the varying soot volume fraction, enabling retrieval of all components. The experimental results show that the reconstructed 2-D scalar fields of temperature [Fig. 9(a)], CO₂ mole fraction [Fig. 9(b)], and soot volume fraction [Fig. 11(c)] in our study are qualitatively consistent with those reported in the literature. A quantitative comparison is conducted for the radial and axial distributions of flame temperature [Fig. 10(a)] and CO₂ mole fraction [Fig. 10(b)]. Although there are deviations between the retrieval results obtained by different methods in the literature and our study, they share a consistent distribution trend. Due to the lack of experimental data, the retrieval results for CO [Fig. 11(a)] and H₂O [Fig. 11(b)] mole fractions are analyzed qualitatively, showing expected trends for the target flame. Additionally, a detailed analysis of the error sources in the reconstruction results is also conducted. Generally, the results confirm the feasibility of adopting the physics-based neural network with hyperspectral measurements to simultaneously reconstruct temperature, multiple gas mole fractions, and soot volume fractions.

We propose a nonlinear tomography reconstruction model based on physics-informed neural networks by combining hyperspectral emission measurements to achieve the simultaneous reconstruction of multiple flame scalar fields. Validation by employing simulated flames shows that the model accurately reconstructs high-resolution scalar fields from mid-infrared spectral data with limited radial projections while maintaining robustness to noise. By adopting experimental data from an ethylene laminar diffusion flame obtained by FTIR, the model successfully retrieves flame temperature, CO₂, H₂O, CO mole fractions, and soot volume fraction. By integrating physical equations directly into the training process, the model eliminates the need for extensive datasets and complex retrieval algorithms. Neural networks exhibit good nonlinear fitting capabilities and the ability to handle high-dimensional data, while also possessing implicit regularization. This allows the neural network to deliver stable and physically reasonable retrieval results. Additionally, as the number of retrieval parameters and the complexity of the problem increase, the problem of difficult retrieval does not significantly intensify, highlighting this model’s potential as a powerful tool for solving complex nonlinear inverse problems.