Estimation of dynamical instability (local Lyapunov exponents) for non-intrusive adaptive modelling

Abstract: 

In chaotic dynamical systems such as the weather, prediction errors grow faster in some situations than in others. Real-time knowledge about the error growth could enable strategies to adjust the forecasting set-up on the fly to increase accuracy and/or reduce computation time. For example one could change the spatio-temporal resolution of the numerical model, locally increase the data availability, etc. Local Lyapunov exponents (LLEs) are known indicators of the rate at which very small prediction errors grow over a finite time interval. However, the classic method of computing LLEs is very expensive, and requires maintaining a tangent linear model. Our work investigates the capability of supervised machine learning to estimate LLEs from recent time steps of the system trajectory, as an alternative to the classical method. Thus machine learning is not used here to emulate (parts of) a physical model, but “non intrusively” as a complementary tool.

We present results from investigations in three low-dimensional chaotic systems of ordinary differential equations: the Rössler and Lorenz 63 systems (three variables) and the one-scale Lorenz 96 system (with 12 variables). In the three-variable systems we perform an extensive investigation of four ML algorithms, using a thorough Bayesian hyperparameter search. We find that the LLE values can only be accurately predicted in certain areas of the attractor. We show that the less accurate predictions occur in areas of the attractor where the LLE values are locally heterogeneous.

In the Lorenz 96 system, our investigations illustrate the challenges of a spatially extended system. We evaluate regression and classification formulations of the ML task. Finally, we investigate approaches for learning which regions of the attractor can be well predicted, so that the LLE predictions are made with a degree of confidence - essential for making decisions in adaptive modelling.