Unsupervised neural-implicit laser absorption tomography for quantitative imaging of unsteady flames

This paper presents a novel neural-implicit approach to laser absorption tomography (LAT) with an experimental demonstration. A coordinate neural network is used to represent thermochemical state variables as continuous functions of space and time. Unlike most existing neural methods for LAT, which rely on prior simulations and supervised training, our approach is based solely on LAT measurements, utilizing a differentiable observation operator with line parameters provided in a standard spectroscopy database format. Although reconstructing scalar fields from multi-beam absorbance data is an inherently ill-posed, nonlinear inverse problem, our continuous space-time parameterization supports physics-inspired regularization strategies and enables data assimilation. Synthetic and experimental tests are conducted to validate the method, demonstrating robust performance and reproducibility. We show that our neural-implicit approach to LAT can capture the dominant spatial modes of an unsteady flame from very sparse measurement data, indicating its potential to reveal combustion instabilities in measurement domains with minimal optical access.

💡 Research Summary

This paper introduces a novel unsupervised neural‑implicit framework for laser absorption tomography (LAT), named NILA‑T (Neural‑Implicit Laser Absorption Tomography). Unlike most recent machine‑learning approaches that rely on supervised training with synthetic data, NILA‑T learns directly from raw LAT measurements without any labeled ground truth. The core idea is to represent the thermochemical state of the gas—mole fraction χ and temperature T—as a continuous function of space and time using a coordinate neural network N : (x, t) → (χ, T). The network is a multilayer perceptron (MLP) augmented with a Fourier positional encoding, which mitigates the low‑frequency bias of standard MLPs and enables the representation of broadband spatial‑temporal features typical of turbulent, unsteady flames.

A differentiable observation operator implements the Beer‑Lambert law together with line‑strength, partition‑function, and lower‑state‑energy data drawn from standard spectroscopy databases (HITRAN/HITEMP). For each laser beam i and each measured transition k, the operator computes a synthetic path‑integrated absorbance by numerically integrating the local absorption coefficient Kₖ(χ,T) along the beam path. Monte‑Carlo sampling approximates the line integral during each training iteration, while automatic differentiation provides exact gradients of the loss with respect to the network parameters.

The total loss consists of three components: (1) a data‑fidelity term J_data that penalizes the L₂ mismatch between measured absorbances Aₖ,i(t) and the synthetic values generated by the network; (2) an explicit regularization term J_reg that applies a second‑order Tikhonov (Laplacian) penalty separately to χ and T, weighted by γ_χ and γ_T; and (3) an optional boundary term J_bound that enforces known ambient conditions on the domain perimeter. The regularization weights are selected via an L‑curve analysis, which balances reconstruction smoothness against data fit and avoids over‑regularization that would erase genuine high‑frequency flame structures.

Synthetic experiments demonstrate that, even with low signal‑to‑noise ratios (≈10 dB) and a sparse set of 30–50 beams, NILA‑T accurately recovers the dominant proper‑orthogonal‑decomposition (POD) modes of a time‑varying flame field. When the Fourier encoding is used without explicit regularization, spurious high‑frequency noise appears; adding the Tikhonov penalty suppresses this noise while preserving the true dynamics.

Experimental validation is performed on a small‑scale burner equipped with a 2‑D LAT array of ~120 beams measuring two CO₂ transitions. Data are acquired at 5 ms intervals over 30 s, providing a time‑resolved dataset for a highly unsteady flame. NILA‑T reconstructs temperature and CO₂ mole‑fraction fields at a temporal resolution of 150 ms, achieving spatial resolution finer than 5 mm. Compared with a conventional algebraic reconstruction technique (ART) with Tikhonov regularization, NILA‑T yields roughly twice the spatial fidelity and markedly improved temporal coherence. The method captures the flame’s dominant oscillation (~20 Hz) and asymmetric structures, and repeated runs show a mean L₂ error below 3.2 % with high reproducibility.

The authors discuss several implications. First, the continuous space‑time representation enables exploitation of spatio‑temporal coherence, effectively acting as a physics‑based prior that reduces the ill‑posedness of the LAT inverse problem. Second, the explicit regularization provides predictable control over smoothness, unlike implicit regularization that depends on network architecture or training dynamics. Third, because the network is differentiable and compact, it can be integrated into data‑assimilation pipelines for real‑time state estimation and uncertainty quantification.

Future work is outlined: extending the framework to include additional state variables such as pressure and velocity; scaling to full 3‑D + t reconstructions for complex combustors; optimizing GPU implementations for near‑real‑time performance; and testing generalization across different fuels and spectroscopic bands.

In summary, NILA‑T demonstrates that an unsupervised neural‑implicit approach, equipped with physics‑based observation models and explicit regularization, can robustly solve the highly ill‑posed LAT problem for unsteady, turbulent flames using only sparse optical access. This opens a pathway toward high‑speed, high‑resolution combustion diagnostics in harsh environments where traditional imaging techniques are impractical.