Machine Learned Empirical Numerical Integrator from Simulated Data

Abstract Recently, a number of state-of-the-art surrogate machine learning (ML) models have been designed for global weather and climate prediction, which have been trained using reanalysis data products. Reanalysis data products are constructed using numerical model simulations that combine numerical integration of partial differential equations and parameterization schemes. These products are typically only archived and made available using coarsened spatial and temporal resolutions. This study explores the impact of the numerical generation methods used to produce the training datasets and the temporal resolution of those datasets on machine learning surrogate models. Using the nonlinear vector autoregression (NVAR) machine as an explainable ML technique, simple dynamical systems are emulated with ML models trained on data produced by three classical numerical integration schemes. NVAR is validated as a skillful ML method, capable of producing accurate predictions and, more importantly, reconstructing both the underlying dynamics and the numerical integration scheme used to generate the training data. However, the machine fails to generalize predictions on unseen test data generated by different numerical integration schemes, despite the underlying dynamical system being the same. This result provides a word of caution for the growing field of machine learning emulation of weather and climate dynamics. Furthermore, we illustrate using NVAR that training on temporally coarsened data may increase the required complexity of ML models and potentially introduce new numerical challenges. Finally, we discover that empirical integration schemes with arbitrary time-stepping sizes can be constructed directly from the data, which implies a potential for the development of empirical numerical integration schemes.