An "Open Box" Physics-Based Neural Network for Modeling Longwave Radiative Transfer

We model longwave radiative transfer using a neural network that assigns physical meaning to the components within its structure. We designate components to represent the radiation sources, spectral decomposition, extinction, scattering, and multi-layer inter-reflection. While we cannot strictly force these components to represent such functionality, we constrain them as much as possible to conform to their assigned functionality. We describe such an approach as an “open box” alternative to the typical black box neural network. It not only exposes various internal variables to interpretation, but also allows us to use physical knowledge to design the network in a manner that facilitates machine learning. We reduce learning complexity by decomposing the physical phenomenon (e.g., longwave radiative transfer) into many smaller components. We then connect the network such that each of these components only receives a necessary but sufficient set of inputs. Furthermore, some components may not need to be learned at all. If accurate physical models exist for these components, we embed these models directly into the neural network structure. For longwave radiative transfer we insert mathematical models for the radiation sources using Planck's law, extinction using Beer's Law, and multi-reflection using the adding algorithm. We then treat the outputs of the components, which correspond to internal nodes of the network, as physical quantities. We structure the network to impose physical laws on these nodes, including conservation of energy, and we incorporate these nodes into the training cost function imposing separate costs on the clear-sky accuracy and full-sky accuracy. We train this system as a single neural network from input to output, jointly learning all trainable weights while constraining the network through the embedded mathematical models and physical laws. Through such a strategy, we developed and trained a compact neural network, with approximately four thousand trainable weights, for modeling longwave radiative transfer. In off-line emulation experiments, this network was trained and validated against the longwave radiative flux and heating rates generated by ecRad (Hogan and Bozzo, 2018), which provides the radiative transfer model for ECMWF's IFS weather forecasting system. Input data was sampled from worldwide reanalysis data from the Copernicus Atmosphere Monitoring Service (CAMS) where each data point included the surface emissivity as well a sixty layer atmospheric profile of temperature, pressure, cloud liquid water, cloud ice water, water vapor, O₃, CO₂, N₂O, CO, O₂, and CH₄. This neural network was trained on data from 2008 and tested on datasets from 2009, 2015, and 2020 with only a small drop-off in accuracy from the 2009 dataset to the 2020 dataset. Overall errors rates for each of these years were no worse than a heating rate root-mean-square-error (RMSE) of 0.11 K day^-1 and a radiative flux RMSE of 0.33 W m^-2