Incorporation of Physical Equations into a Neural Network Structure for Predicting Shortwave Radiative Heat Transfer

We describe an approach that incorporates the equations of physical processes into a neural network structure whereby these equations represent known physical laws and where multilayer perceptron modules (MLPs) represent physical processes that are difficult to express analytically. This structure seeks to enforce physical laws within a data-driven approach making efficient use of neural network capacity and training data. This structure also allows for physical interpretations of various network nodes and network flows. We demonstrate this approach for prediction of shortwave radiative heat transfer. We hard code the equations for Beer's law and the adding-doubling method. MLPs compute the spectral decomposition, scattering, and the dependence of the mass extinction coefficients on temperature and pressure. Specific network nodes may be interpreted as optical depths and the coefficients of transmission, reflection, and absorption at individual atmospheric layers. Pathways in the network may be interpreted as components of the direct and diffuse radiative fluxes. We train a single network structure with approximately 1,200 weights, incorporating the entirety of these hard-coded physical laws and MLPs. The training loss function includes errors in flux and errors in heating rates computed from the downwelling direct radiative flux and the total downwelling and upwelling radiative fluxes. In off-line emulation experiments, this system was trained and validated against the radiative flux and heating rates generated by RTE+RRTMGP (Pincus, Mlawer, and Delamere, 2019). Input data was sampled from several years of worldwide reanalysis data from the Copernicus Atmosphere Monitoring Service (CAMS) where each data point included the solar zenith angle and surface albedo as well a 60 layer atmospheric profile of temperature, pressure, water vapor, liquid water, ice water, O₃, CO₂, N₂O, and CH₄. On testing data which was collected over a time period that was disjoint from the training data, the system had a heating rate root-mean-square-error (RMSE) of less than 0.18 K day^-1 and a flux RMSE of less than 3 W m^-2.