Neural Networks 21 (2008) 535–543 www.elsevier.com/locate/neunet 2008 Special Issue Neural network approach for robust and fast calculation of physical processes in numerical environmental models: Compound parameterization with a quality control of larger errors ✩, ✩✩ Vladimir M. Krasnopolsky a,b,∗ , Michael S. Fox-Rabinovitz b , Hendrik L. Tolman c , Alexei A. Belochitski b a Science Applications International Corporation at Environmental Modeling Center, National Centers for Environmental Prediction, National Oceanic and Atmospheric Administration, MD, USA b Earth System Science Interdisciplinary Center, University of Maryland, MD, USA c Environmental Modeling Center, National Centers for Environmental Prediction, National Oceanic and Atmospheric Administration, MD, USA Received 2 August 2007; received in revised form 13 November 2007; accepted 13 December 2007 Abstract Development of neural network (NN) emulations for fast calculations of physical processes in numerical climate and weather prediction models depends significantly on our ability to generate a representative training set. Owing to the high dimensionality of the NN input vector which is of the order of several hundreds or more, it is rather difficult to cover the entire domain, especially its “far corners” associated with rare events, even when we use model simulated data for the NN training. Moreover the domain may evolve (e.g., due to climate change). In this situation the emulating NN may be forced to extrapolate beyond its generalization ability and may lead to larger errors in NN outputs. A new technique, a compound parameterization, has been developed to address this problem and to make the NN emulation approach more suitable for long-term climate prediction and climate change projections and other numerical modeling applications. Two different designs of the compound parameterization are presented and discussed. c  2008 Elsevier Ltd. All rights reserved. Keywords: Neural networks; Numerical modeling; Climate; Weather; Waves; Data assimilation 1. Introduction This paper describes an interdisciplinary study. This study follows upon our previous works presented in the previous papers of the authors (e.g., Krasnopolsky, Chalikov, and Tolman (2002) and Krasnopolsky, Fox-Rabinovitz, and Chalikov (2005)). In these works we developed a new approach, introducing nonlinear statistical learning techniques (NNs) into tremendously complex and time consuming numerical models, describing one of the most complex, multidimensional, ✩ MMAB Contribution No. 258. ✩✩ An abbreviated version of some portions of this article appeared in Krasnopolsky, Fox-Rabinovitz, and Belochitski (2007) as part of the IJCNN 2007 Conference Proceedings, published under IEE copyright. ∗ Tel.: +1 301 763 8000; fax: +1 301 763 8545. E-mail address: vladimir.krasnopolsky@noaa.gov (V.M. Krasnopolsky). and essentially nonlinear systems (climate/weather system) known to the modern science. This new approach introduces fast and accurate NN emulations of time consuming original model components into numerical climate/weather models. As a result, the model computational performance improves significantly without a detriment to the quality of model predictions. This applied research (and the current study) has a clearly formulated practical goal: to improve computational performance of operational weather prediction and climate simulation models by using accurate, fast, and robust NN emulations substituting the time consuming original components of the models. 1.1. Climate models and model physics One of the main problems of development and implemen- tation of high-quality high-resolution environmental models is the complexity of physical (chemical and biological) processes 0893-6080/$ - see front matter c  2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.neunet.2007.12.019