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 modelling and forecasting infrastructure on the fly to increase accuracy and/or reduce computation time. For example, one could change the ensemble size, the distribution and type of target observations, and so forth. Local Lyapunov exponents are known indicators of the rate at which very small prediction errors grow over a finite time interval. However, their computation is very expensive: it requires maintaining and evolving a tangent linear model, orthogonalisation algorithms and storing large matrices. In this feasibility study, we investigate the accuracy of supervised machine learning in estimating the current local Lyapunov exponents, from input of current and recent time steps of the system trajectory, as an alternative to the classical method. Thus machine learning is not used here to emulate a physical model or some of its components, but “nonintrusively” as a complementary tool. We test four popular supervised learning algorithms: regression trees, multilayer perceptrons, convolutional neural networks, and long short‐term memory networks. Experiments are conducted on two low‐dimensional chaotic systems of ordinary differential equations, the Rössler and Lorenz 63 models. We find that on average the machine learning algorithms predict the stable local Lyapunov exponent accurately, the unstable exponent reasonably accurately, and the neutral exponent only somewhat accurately. We show that greater prediction accuracy is associated with local homogeneity of the local Lyapunov exponents on the system attractor. Importantly, the situations in which (forecast) errors grow fastest are not necessarily the same as those in which it is more difficult to predict local Lyapunov exponents with machine learning.

<p>Weather and climate are well known exemplars of chaotic systems exhibiting extreme sensitivity to initial conditions. Initial condition errors are subject to exponential growth on average, but the rate and the characteristic of such growth is highly state dependent. In an ideal setting where the degree of predictability of the system is known in real-time, it may be possible and beneficial to take adaptive measures. For instance a local decrease of predictability may be counteracted by increasing the time- or space-resolution of the model computation or the ensemble size in the context of ensemble-based data assimilation or probabilistic forecasting.</p><p>Local Lyapunov exponents (LLEs) describe growth rates along a finite-time section of a system trajectory. This makes the LLEs the ideal quantities to measure the local degree of predictability, yet a main bottleneck for their real-time use in  operational scenarios is the huge computational cost. Calculating LLEs involves computing a long trajectory of the system, propagating perturbations with the tangent linear model, and repeatedly orthogonalising them. We investigate if machine learning (ML) methods can estimate the LLEs based only on information from the system’s solution, thus avoiding the need to evolve perturbations via the tangent linear model. We test the ability of four algorithms (regression tree, multilayer perceptron, convolutional neural network and long short-term memory network) to perform this task in two prototypical low dimensional chaotic dynamical systems. Our results suggest that the accuracy of the ML predictions is highly dependent upon the nature of the distribution of the LLE values in phase space: large prediction errors occur in regions of the attractor where the LLE values are highly non-smooth.  In line with classical dynamical systems studies, the neutral LLE is more difficult to predict. We show that a comparatively simple regression tree can achieve performance that is similar to sophisticated neural networks, and that the success of ML strategies for exploiting the temporal structure of data depends on the system dynamics.</p>