Estimating the Global, Beam and Diffuse Components of Photosynthetically Active Radiation Using Physically Constrained Neural Network

Surface photosynthetically active radiation (PAR; 400–700 nm) components—global horizontal, beam normal, and diffuse horizontal—are critical for ecosystem and solar energy applications, particularly in open-field agrivoltaic systems. However, the scarcity of outdoor measurements limits the availability of these data. Machine learning (ML) offers a powerful and practical alternative to traditional parameterization-based methods to modeling the three PAR components. This study develops a physically constrained multi-output neural network based on the multi-gate mixture-of-experts (MMoE) architecture to simultaneously estimate the three PAR components. A specially designed constraint layer ensures adherence to the fundamental physical relationship (i.e., the closure equation). Compared to several single-output semi-empirical and ML models, the proposed model demonstrates superior performance on the test set, with average normalized root mean square error (nRMSE), normalized mean absolute error (nMAE), normalized mean bias error (nMBE), and R<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> values of 4.1997%, 2.8888%, -0.9775%, and 0.9975, respectively. Notably, the model's outputs perfectly satisfy the physical constraint, achieving zero relevant errors, whereas other ML models exhibit varying degrees of errors, with their maximum values ranging from 1% to 3%.