A nonintrusive hybrid neural-physics modeling of incomplete dynamical systems: Lorenz equations

Abstract

This work presents a hybrid modeling approach to data-driven learning and representation of unknown physical processes and closure parameterizations. These hybrid models are suitable for situations where the mechanistic description of dynamics of some variables is unknown, but reasonably accurate observational data can be obtained for the evolution of the state of the system. In this work, we propose machine learning to account for missing physics and then data assimilation to correct the prediction. In particular, we devise an effective methodology based on a recurrent neural network to model the unknown dynamics. A long short-term memory (LSTM) based correction term is added to the predictive model in order to take into account hidden physics. Since LSTM introduces a black-box approach for the unknown part of the model, we investigate whether the proposed hybrid neural-physical model can be further corrected through a sequential data assimilation step. We apply this framework to the weakly nonlinear Lorenz model that displays quasiperiodic oscillations, the highly nonlinear Lorenz model, and two-scale Lorenz model. The hybrid neural-physics model yields accurate results for the weakly nonlinear Lorenz model with the predicted state close to the true Lorenz model trajectory. For the highly nonlinear Lorenz model and the two-scale Lorenz model, the hybrid neural-physics model deviates from the true state due to the accumulation of prediction error from one time step to the next time step. The ensemble Kalman filter approach takes into account the prediction error and updates the diverged prediction using available observations in order to provide a more accurate state estimate. The successful synergistic integration of neural network and data assimilation for low-dimensional system shows the potential benefits of the proposed hybrid-neural physics model for complex systems.