physics.opticsOct 17, 2026

Enhancing analytical photon transport in turbid media through physics-guided neural network corrections

Authors: Guojun Tang, Paul D. Teal, Alexander Doronin

Organizations: Victoria University of Wellington, School of Engineering and Computer Science, Wellington, New Zealand

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Abstract

Analytical methods provide rapid and computationally efficient predictions of light transport in turbid media, making them attractive for real-time and inverse applications. However, their accuracy is limited by the approximations and assumptions used to obtain tractable solutions. On the other hand, Monte Carlo (MC) simulation is widely regarded as the "gold standard" for modeling light transport in turbid media, but its high computational cost limits its use in applications requiring repeated forward evaluations. Here, we introduce a physics-guided residual learning framework that combines the computational efficiency of analytical modeling with the accuracy of MC simulation. As a representative case study, we consider the diffusion approximation (DA), which can exhibit systematic errors under relatively strong absorption, weak scattering, and imperfect boundary treatment. Given the absorption coefficient, scattering coefficient, and anisotropy factor, the corresponding DA prediction is incorporated as a physical prior, while the network learns only the residual between the analytical prediction and the MC reference. Across the evaluated optical-property range, the proposed framework reduces scalar-reflectance RMSE and spatially resolved reflectance RMSLE by factors of approximately 62 and 14, respectively, relative to the diffusion approximation. With only 1% of the available training data, it achieves approximately 87% and 93% lower corresponding errors than a direct neural-network surrogate. These results demonstrate that learning corrections to an analytical physical baseline improves both prediction accuracy and data efficiency.

Figures & tables

Figure 1: (a) Spatially resolved diffuse reflectance Rd(ρ)R_{\text{d}}(\rho) in a medium with optical properties (μa,μs,g)(\mu_{\text{a}},\mu_{\text{s}},g) . (b) Conceptual illustration of the discrepancy between the diffusion approximation (dashed) and Monte Carlo (solid). The shaded gap, ΔRd(ρ)\Delta R_{\text{d}}(\rho) , represents the correction learned by the proposed framework.
Figure 2: The Conceptual Pipeline of the Physical-Guided Correcting Diffusion Approximation.
Figure 3: The Proposed Physics-Guided Residual Learning architecture
ModelRMSER2R^{2}
Farrell DA1.77×10−21.77\times 10^{-2}0.988689
MLP(5.20±0.28)×10−4(5.20\pm 0.28)\times 10^{-4}0.999990
PGRL(2.85±0.05)×10−4(2.85\pm 0.05)\times 10^{-4}0.999997
Table 1: Prediction performance for scalar diffuse reflectance using the full training set.
ModelRMSER2R^{2}
Farrell DA1.77×10−21.77\times 10^{-2}0.988689
MLP(4.17±0.41)×10−3(4.17\pm 0.41)\times 10^{-3}0.999368±0.0001260.999368\pm 0.000126
MLP + Farrell Input(1.13±0.07)×10−3(1.13\pm 0.07)\times 10^{-3}0.999954±0.0000060.999954\pm 0.000006
Residual MLP(1.17±0.28)×10−3(1.17\pm 0.28)\times 10^{-3}0.999948±0.0000260.999948\pm 0.000026
PGRL(0.554±0.15)×10−3(0.554\pm 0.15)\times 10^{-3}0.999989±0.0000010.999989\pm 0.000001
Table 2: Ablation study of the proposed physics-guided components for scalar diffuse reflectance using 1% of the training data.
Figure 4: Performance of the proposed physics-guided residual learning (PGRL) framework for scalar and spatially resolved diffuse reflectance. (a) Test-set RMSE of the direct MLP and PGRL models as the amount of available MC training data is varied. (b) Comparison between predicted scalar diffuse reflectance RdR_{\mathrm{d}} and the MC reference across the investigated optical-property space. (c) Spatially resolved reflectance profiles Rd(ρ)R_{\mathrm{d}}(\rho) for μa=10\penalty cm−1\mu_{\mathrm{a}}=10\penalty\ \mathrm{cm}^{-1} , μs=90\penalty cm−1\mu_{\mathrm{s}}=90\penalty\ \mathrm{cm}^{-1} , and g=0g=0 , 0.50.5 , and 0.90.9 . (d) Corresponding two-dimensional radially symmetric reflectance maps. The PGRL model corrects systematic deviations of the Farrell diffusion approximation while retaining the smooth spatial structure of the MC reference.
ModelRMSLER2R^{2}
Farrell DA0.2022400.2022400.9641550.964155
MLP0.233±0.2300.233\pm 0.2300.997102±0.0062870.997102\pm 0.006287
PGRL0.014±0.0020.014\pm 0.0020.999890±0.0000290.999890\pm 0.000029
Table 3: Prediction performance for spatially resolved diffuse reflectance using the full training set.
ModelRMSLER2R^{2}
Farrell DA0.2022400.2022400.9641550.964155
MLP0.454±0.3130.454\pm 0.3130.593854±0.0513090.593854\pm 0.051309
MLP + Farrell Input0.427±0.2770.427\pm 0.2770.658110±0.1029530.658110\pm 0.102953
Residual MLP0.049±0.0030.049\pm 0.0030.975164±0.0007460.975164\pm 0.000746
PGRL0.033±0.0030.033\pm 0.0030.977060±0.0010080.977060\pm 0.001008
Table 4: Ablation study of the proposed physics-guided components for spatially resolved diffuse reflectance using 1% of the training data.
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