Linearized solutions of the scalar radiative transfer equation using physics-informed neural networks

Fast and accurate solutions of the radiative transfer equation and its linearization are fundamental to atmospheric remote sensing. Jacobians of radiative quantities with respect to atmospheric and surface parameters are essential for sensitivity analysis and iterative retrieval. However, computing them with conventional linearized radiative transfer models is computationally expensive. In this study, we develop a physics-informed radiative transfer neural network (PIRTNN) for the efficient approximation of shortwave radiative fields and their associated Jacobians. The framework is formulated for a homogeneous, plane-parallel, single-layer scattering atmosphere over a Lambertian surface and is trained, validated, and tested using forward and analytical linearized solutions generated by the linearized discrete ordinate radiative transfer (LDISORT) model. The neural network takes total optical depth, single scattering albedo, asymmetric factor, surface albedo, solar zenith angle, and azimuth angle as input features, and outputs include viewing geometry-dependent radiance, upward and downward fluxes, actinic flux, and the corresponding Jacobians with respect to input parameters, excluding solar geometry variables. Model performance is evaluated in terms of accuracy, distributions, and parameter sensitivities. Results demonstrate that the PIRTNN reproduces LDISORT-computed radiative quantities and Jacobians with high accuracy within the considered parameter space. On a GPU, PIRTNN performs simultaneous prediction of radiances, fluxes, and their derivatives with respect to four physical parameters with a batch size of 10⁵ samples in 4.85 s. These results demonstrate the feasibility of using physics-informed neural networks to approximate both the forward and linearized solutions of the radiative transfer equation and provide a basis for extending the framework to vertically inhomogeneous multilayer atmospheres.