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. Despite the great success of machine learning, its application in climatedynamics has not been well developed. One concern might be how well thetrained neural networks could learn a dynamical system and what will be thepotential application of this kind of learning. In this paper, three machine-learning methods are used: reservoir computer (RC), backpropagation-based (BP)artificial neural network, and long short-term memory (LSTM) neural network. It shows that the coupling relations or dynamics among variables inlinear or nonlinear systems can be inferred by RC and LSTM, which can befurther applied to reconstruct one time series from the other. Specifically,we analyzed the climatic toy models to address two questions: (i) whatfactors significantly influence machine-learning reconstruction and (ii)how do we select suitable explanatory variables for machine-learningreconstruction. The results reveal that both linear and nonlinear couplingrelations between variables do influence the reconstruction quality ofmachine learning. If there is a strong linear coupling between twovariables, then the reconstruction can be bidirectional, and both of thesetwo variables can be an explanatory variable for reconstructing the other.When the linear coupling among variables is absent but with the significantnonlinear coupling, the machine-learning reconstruction between twovariables is direction dependent, and it may be only unidirectional. Thenthe convergent cross mapping (CCM) causality index is proposed to determinewhich variable can be taken as the reconstructed one and which as theexplanatory variable. In a real-world example, the Pearson correlationbetween the average tropical surface air temperature (TSAT) and the averageNorthern Hemisphere SAT (NHSAT) is weak (0.08), but the CCM index of NHSATcross mapped with TSAT is large (0.70). And this indicates that TSAT can be wellreconstructed from NHSAT through machine learning. All results shown in this study could provide insights into machine-learningapproaches for paleoclimate reconstruction, parameterization scheme, andprediction in related climate research. Highlights: i The coupling dynamics learned by machine learning can be used to reconstructtime series. ii Reconstruction quality is direction dependent and variable dependent for nonlinearsystems. iii The CCM index is a potential indicator to choose reconstructed andexplanatory variables. iv The tropical average SAT can be well reconstructed from the average NorthernHemisphere SAT.

Reconstructing coupled time series in climate systems using three kinds of machine-learning methods