The El Niño–Southern Oscillation (ENSO) can induce global climate extremes and ecosystem impacts (Zhang et al., 2016), which are the dominant sources of interannual climate changes. The El Niño (La Niña) is the ocean phenomena of ENSO and is usually considered as the large-scale positive (negative) sea surface temperature (SST) anomalies in the tropical Pacific Ocean. The Niño 3 (Niño 4) index is the common indicator for ENSO research to measure the cold tongue (warm pool) variabilities, which is the averaged SST anomalies covering the Niño 3 (Niño 4) region (see Fig. 1). Besides these two indicators, the ONI (oceanic Niño index, 3-month running mean of SST anomalies in the Niño 3.4 region) has become the de facto standard to identify the occurrence of El Niño and La Niña events: if the ONIs of 5 consecutive months are over 0.5 ∘C (below -0.5 ∘C), El Niño (La Niña) occurs.

Conventional forecast approaches mainly rely on numerical climate models. However, it is worth noting that the model biases of traditional approach have always been a problem for accurate ENSO predictions (Xue et al., 2013). In addition, many other intrinsic factors also limit the ENSO predictability such as natural decadal variations in ENSO amplitudes. For example, predictability tends to be higher when the ENSO cycle is strong than when it is weak (Barnston et al., 2012; Balmaseda et al., 1995; McPhaden, 2012). Recently, due to deluges of multi-source real-world geoscience data starting to accumulate, e.g., remote sensing and buoy observation, meteorological researchers were inspired to build lightweight and convenient data-driven models at a low computational cost (Rolnick et al., 2019), which lead to a wave of formulating ENSO forecast with deep learning techniques, producing more skilful ENSO predictions (Ham et al., 2019).

In the field of deep learning, ENSO prediction is usually regarded as forecasting the future evolution tendency of SST and related Niño indexes directly, subsequently analyzing the associated sophisticated mechanisms, and measuring the intrinsic characteristics such as intensity and duration. Therefore, the simplest but most practical forecast manners can be divided into two categories intuitively: Niño index forecast and SST pattern forecast.

As for Niño index forecasting, many favorable neural networks have made accurate predictions 6, 9 and 12 months ahead. For instance, ensemble QESN (McDermott and Wikle, 2017), BAST-RNN (McDermott and Wikle, 2019) and LSTM (long short-term memory) (Broni-Bedaiko et al., 2019) are representative works. These studies demonstrate that the deep learning can well capture the nonlinear characteristics of non-stationary time series and attain outstanding regressions on Niño index.

Notwithstanding the successful attempts on the Niño index regression, there still exist many pitfalls in measuring ENSO forecast skills by only one single scalar. For example, the important spatial–temporal energy propagations and teleconnections cannot be described by the indexes. It may lead to the blind pursuit of the accuracy of a certain indicator while seriously hampering the grasp of underlying physical mechanisms. Therefore, many studies are suggestive of exploiting spatial–temporal dependencies and predicting the evolution of SST patterns. Ham et al. (2019) apply transfer learning (Yosinski et al., 2014) to historical simulations from CMIP5 (Coupled Model Intercomparison Project phase5, Bellenger et al., 2014) and reanalysis data with a CNN model to predict ENSO events, resulting in a robust and long-term forecast for up to 1.5 years, which outperforms the current numerical predictions. (Though the output of their model is still the Niño 3.4 index, they construct the model and make forecasts by absorbing the historical spatial–temporal features from variable patterns instead of previous index records, so we mark this study as SST pattern forecasts in this paper.) Mu et al. (2019) and He et al. (2019) built a ConvLSTM (Shi et al., 2015) model to capture the spatial–temporal dependencies of ENSO SST patterns over multiple time horizons and obtained better predictions. Zheng et al. (2020) constructed a purely satellite-data-driven deep learning model to forecast the evolutions of tropical instability wave, which is closely related to ENSO phenomena, and obtained accurate and efficient forecasts. These deep learning models tend to simulate the behaviors of numerical climate models, the inputs of which are historical geoscience data and the outputs of which are the forecasted SST patterns.

The reason for the great progress in these works is no accident. On the one hand, the deep learning models have much more complex structures and can mine the complicated features hidden in the samples more effectively, which allows them to be substantially more expressive with blending the non-stationarity in temporal and the multi-scale teleconnections in spatial. On the other hand, it is very convenient to migrate deep learning computer vision technologies to ENSO forecasting due to the nature analogy between the format of image/video frame data and meteorological time-series grid data, which offers promises for extracting spatial–temporal mechanisms of ENSO via advanced deep learning techniques. Therefore, the data-driven deep learning can be a reliable alternative to traditional numerical models and a powerful tool for the ENSO forecasting.

However, there are still some obstacles in the deep learning modeling process for ENSO forecasting. Very often, most existing models are confined to limited or even single input predictors, such as only using historical SST (and wind) data as the model input. Meanwhile, the climate deep learning models are rarely adaptively customized to the specific physical mechanisms of ENSO. These situations lead to poor interpretability and low confidence of ENSO-related deep learning models. ENSO is an extremely complicated ocean–atmosphere coupling event, and the development and decay phases are closely associated with some crucial dynamic mechanisms and Walker circulation (Bayr et al., 2020), whose status have great impacts. Walker circulation is usually modulated by multi-physical variables (such as SST, wind, precipitation, etc.), and there are always coupling interactions between different variables. More specifically, the varieties of the Walker circulation have strong temporal-lag effects on ENSO (“memory effects”). The position of the ascending branch is also a very important climatic condition for the occurrence of El Niño. Such a priori ENSO knowledge has not been effectively used in deep learning model.

Therefore, in order to further improve the ENSO prediction skill, there is an essential principle that should be reflected in climate deep learning models: subjectively incorporating the a priori ENSO knowledge into the deep learning formalization and deriving hand-crafted features to make predictions.

In this paper, according to the important synergies of multiple variables in crucial ENSO dynamic mechanisms and Walker circulation, we select six indispensable variables (SST, u wind, v wind, rain, cloud, and vapor) that are induced from ENSO-related key processes to build a multivariate air–sea coupler (ASC) based on a graph mathematically, which emphasizes the energy exchange between multiple variables. We then leverage this coupler to build up the ENSO deep learning forecast model, named ENSO-ASC, with an encoder–coupler–decoder structure to extract the multi-scale spatial–temporal features of multiple physical variables. Two attention weights are also proposed to grasp the different air–sea coupling strengths on different start calendar months and varied effects of these variables. A loss function combining MSE (mean squared error) and MAE (mean absolute error) is used to guide the model training precisely, and SSIM (structural similarity) (Wang et al., 2004) and PSNR (peak signal-to-noise ratio) are used as metrics to evaluate the spatial consistency of the forecasted patterns.

Two datasets are applied for model training to ensure that the systematic forecast errors are fully corrected after tuning by the higher quality dataset: we first train the ENSO-ASC on the numerous reanalysis samples from January 1850 to December 2015 and subsequently on the high-quality remote sensing samples from December 1997 to December 2012 for fine-tuning. This procedure is also known as transfer learning. These two datasets are modulated to the same resolution. The validation period is from January 2014 to August 2020 in the remote sensing dataset. The gap between the fine-tuning set and validation set is used to remove the possible influence of oceanic memory (Ham et al., 2019).

This is the first time that a multivariate air–sea coupler has been designed that considers energy interactions. We evaluate the ENSO-ASC from three aspects: firstly, we evaluate the model performance from the perspective of model structure, including the input sequence length, the benefits of transfer learning, multivariate air–sea coupler, and the attention weights, and tune the model structure to optimal. Then, we analyze the ENSO forecast skill of the ENSO-ASC from the meteorological aspects, including the contributions of different input physical variables, the effectiveness of forecast lead time, the forecast skill changes with different start calendar months, and the forecast spatial uncertainties. Subsequently, we make the real-world ENSO simulations during the validation period by tracing the evolutions of multiple physical variables. From the experiment results, ENSO-ASC performs better in both SSIM and PSNR of the forecasted SST patterns, which effectively raises the upper limitation of ENSO forecasts. The forecasted ENSO events are more consistent with real-world observations and the related Niño indexes have higher correlations with observations than traditional methods and current state-of-the-art deep learning models, which is over 0.78/0.65/0.5 within the lead time of 6/12/18 months for Niño 3.4 index. SST and zonal wind are two crucial predictors, which can be considered as the major triggers of ENSO. A temporal heat map analysis illustrates that the ENSO forecasting skills decline faster when making forecasts through June–July–August, and a spatial heat map analysis shows that the forecast errors are more likely to show up over the central tropical Pacific Ocean in longer-term forecasts. Meanwhile, in the validation period from 2014 to 2020, the multivariate air–sea coupler can capture the latent ENSO dynamical mechanisms and provide multivariate evolution simulations with a high degree of physical consistency: The positive SST anomalies first show up over the eastern equatorial Pacific with the westerly wind anomalies in the western and central tropical Pacific Ocean (vice versa in the La Niña events), which induces Bjerknes positive feedback mechanism. It is worth noting that for the simulation of the 2015–2016 super El Niño, ENSO-ASC captures its strong evolutions of SST anomalies over the northeast subtropical Pacific in the peak phase and successfully predicts its very-high-intensity and very-long-duration, while many dynamic or statistical models fail. At the same time, ENSO-ASC can also reduce false alarm rate such as in 2014. From the mathematical expression, the multivariate air–sea coupler captures the spatial–temporal multi-scale oscillations of the Walker circulation and performs the ocean–atmosphere energy exchange simultaneously, which tries to avoid the interval flux exchange in geoscience fluid programming of traditional numerical climate models. In conclusion, the graph-based multivariate air–sea coupler not only exhibits effectiveness and superiority to predict sophisticated climate phenomena, but is also a promising tool for exploiting the underlying dynamic mechanisms in the future.

The remainder of this paper is organized as follows. Section 2 introduces the proposed multivariate air–sea coupler. Section 3 describes the ENSO deep learning forecast model with the coupler (ENSO-ASC) in detail. Section 4 illustrates the datasets, experiment schemas and result analyses. Finally, Sect. 5 offers further discussions and summarizes the paper.

ENSO is the most dominant phenomenon of air–sea coupling over the equatorial Pacific, and many complex dynamical mechanisms modulate the ENSO amplitudes. Bjerknes positive feedback (Bjerknes, 1969) is one of the most significant effects, the processes of which are highly related to the status of the Walker circulation. There are energy interactions between the multiple physical variables influenced by Walker circulation every moment, and the ENSO-related SST varieties are greatly affected by such air–sea coupling activities (Gao and Zhang, 2017; Lau et al., 1989; Lau et al., 1996).

Many atmospheric and oceanic anomalies are known as triggers of ENSO events, which establish the Bjerknes positive feedback. The warming SST anomalies propagate to the central and eastern equatorial Pacific gradually. As SST gradually rises, it is virtually impossible for the equatorial Pacific to enter a never-ending warm state. Therefore, some negative feedback will cause turnabouts from warm phases to cold phases (Wang et al., 2017). These negative feedback mechanisms all emphasize air–sea interactions. For example, westerly wind anomalies in the central tropical Pacific Ocean induce the upwelling Rossby and downwelling Kevin oceanic waves, both of which propagate and reflect on the continental boundary and then tend to push the warm pool back to its original position in the western Pacific. From the perspective of ENSO life cycle, atmospheric and oceanic variables play crucial roles together.

Meanwhile, during the development and decay phases of ENSO, there also exist nonlinear interactions between atmospheric and oceanic variables. Wind anomalies are the most obvious and direct response of the ENSO-driven large-scale oceanic varieties, and they will change the ocean–atmosphere heat transmissions (Cheng et al., 2019). Once the ocean status changes, the thermal energy contained in the sea will escalate or dissipate into the air, hindering or promoting the precipitation and surface humidity over the equatorial Pacific. These changes also give feedback on the ENSO.

Meteorological researchers have already identified the key physical processes in ENSO in recent years. If such knowledge can be incorporated into ENSO deep learning forecast modeling subjectively, breaking away from the current limitation of using single predictors, the accuracy of ENSO prediction will promise breakthroughs. In this paper, we choose six ENSO-related indispensable variables from two different multivariate datasets as shown in Table 1, which all have strong correlations within the evolution of ENSO events according to Bjerknes positive feedback and other dynamical processes. Furthermore, in order to comprehensively represent the coupling interactions, a multivariate air–sea coupler coupler(G) is designed to simulate their synergies with an undirected graph G=(V,A) as shown in Fig. 2, where V=(fv1fv2,…,fvN) represents the vertices of the graph and fvi is the feature of every physical variable vi (i=1,2,…,N). A∈RN×N is the pre-designed adjacency matrix, where Ai,j=1 (Ai,j=0) represents the existing (non-existent) energy interactions between the connected variables vi and vj. The variables exchange energies simultaneously every moment, and the directions of edges in this graph can be neglected because the energy interactions are two-way (transfer and feedback).

Here, V in G is not the physical variable on a single grid point, but the features of the entire variable pattern. The reason lies in the following: on the one hand, the coupler will pay more attention to the global and local spatial–temporal correlations in the variable fields of ENSO rather than the variations on an isolated grid. On the other hand, the coupler will provide a higher computational efficiency and consume a lower calculation resource for ENSO forecasting. Improvements to the couplers, such as designing individual graph for smaller-scale regions and even a single grid, are future considerations.

Inspired by previous ENSO deep learning forecast models, we can define the ENSO forecast as a multivariate spatial–temporal sequence forecast problem as illustrated in Eq. (1), 1s^=Fθ(sscm),{sst,uwind,vwind,rain,cloud,vapor}⊆sscm, where sscm∈RN×M is N multivariate observations in historical M months (N=6), and s^∈RN×H is the prediction result for future H months (H can be also treated as forecast lead time). scm∈{Jan,Feb,…,Dec} (start calendar month) represents the last month in the input series sscm. Fθ represents the forecast system (θ denotes the trainable parameters in the system).

In order to incorporate the multivariate coupler, we break down the conventional formulation and redefine the multivariate ENSO forecast model as the encoder–coupler–decoder structure shown as in Eqs. (2) to (4), 2encoder:fvi=encoderi(siscm),3coupler:fc=couplerG=coupler((fv1||fv2||…||fvN),A),4decoder:s^i=decoderi(fvi||fci), where the siscm(i=1,2,…,N) represents the individual physical data and fvi represents the corresponding extracted features by respective encoders. The coupler (⚫) simulates the latent multivariate interactions on the physical features and the pre-designed interaction graph A, where the operator || represents the concatenation of features of different physical variables. Then the respective decoders will restore the physical features end-to-end by the coupled multivariate features fci and original physical features fvi, the concatenation of which can be regarded as skip-layer connections. These connections can propagate the low-level feature to high levels of the model directly, preserving the raw information and accelerating feature transfers to some extent. The sub-modules encoderi(⚫), coupler(⚫), and decoderi(⚫) form the ENSO-ASC together.

As mentioned before, the strength of the multivariate coupling and the effects of multivariate temporal memories in ENSO are changing with different input sequence length M and forecast start calendar month scm. In order to grasp such effects on forecasts, we design two self-supervised attention weights, α=(a1,a2,…,aM) and β=(b1,b2,…,bN), in the encoder and coupler respectively to capture the dynamic time-series non-stationarity and re-weight the multivariate contributions. The final formulation of the forecast model can be written as shown in Eqs. (5) to (7) and in Fig. 3, where ∘ represents the element-wise multiplication. 5encoder:fvi=α∘encoderi(siscm)6coupler:fc=couplerβ∘G=coupler((b1fv1||b2fv2||…||bNfvN),A)7decoder:s^i=decoderi(fvi||fci)

In addition, there are basically two forecast strategies for ENSO prediction: direct multi-step (DMS) and iterated multi-step (IMS) (Chevillon, 2007). The former means predicting the future Hth-month multivariate pattern directly, and the latter means utilizing the forecast output result as the input for future iterated predictions. Figure 4 displays the differences between DMS and IMS. In general, DMS is often unstable and more difficult to train for a deep learning model (Shi and Yeung, 2018). Therefore, we choose IMS to handle chaos data and provide more accuracy predictions, that is, forecasting the next 1-month (H=1) multivariate data in the model, and then using this output as model input to continuously predict the future evolutions. We also design a combined loss function to train our model and use two spatial metrics to evaluate the forecast results. The intentions and detailed implementations of every part in the model are interpreted as the following sections.

ENSO is a very complicated air–sea coupled phenomenon, the life cycle of which is closely related to the large-scale nonlinear interactions between various oceanic and atmospheric processes. ENSO is one of the most critical factors that cause extreme climatic and socioeconomic effects. Therefore, meteorological researchers are starting to find more accurate and less consuming data-driven models to forecast ENSO, especially deep learning methods. There are already many successful attempts that have extended the effective forecast lead time of ENSO up to 1.5 years. They all extract the rich spatial–temporal information deeply hidden in the historical geoscience data.

However, most of the models use limited variables or even a single variable to predict ENSO, ignoring the coupling multivariate interactions in ENSO events. At the same time, the generic ENSO deep learning forecast models seem to have reached the performance bottleneck, which means deeper or more complex model structures can neither extend the effective forecast lead time nor provide a more detailed description for dynamical evolutions. In order to overcome these two barriers, we subjectively incorporate a priori ENSO knowledge into the deep learning formalization and derive hand-crafted features into models to make predictions. More specifically, considering the multivariate coupling in the Walker circulation related to ENSO amplitudes, we select six indispensable physical variables and focus on the synergies between them in ENSO events. Instead of simple variable stacking, we treat them as separate individuals and ingeniously formulate the nonlinear interactions between them on a graph. Based on such formalization, we design a multivariate air–sea coupler (ASC) by graph convolution mathematically, which can mine the coupling interactions between every physical variable in pairs and perform the multivariate synergies simultaneously.

We then implement an ENSO deep learning forecast model (ENSO-ASC) with the encoder–coupler–decoder structure, and two self-supervised attention weights are also designed. The multivariate time-series data are firstly propagated to the encoder to extract spatial–temporal features respectively. Then the multivariate features are aggregated together for interactions in the multivariate air–sea coupler. Finally, the coupled features are divided separately, and the corresponding feature of a certain variable is restored to forecast patterns in the decoder. IMS strategy is applied to make predictions, which is a more stable forecasting method. We use transfer learning to provide a better model initialization and overcome the problem of a lack of observation samples. The model is first trained on the reanalysis dataset and subsequently on the remote sensing dataset. After constructing the model structure, we design extensive experiments to investigate the model performance and ENSO forecast skill. Several successful simulations in the validation period are also provided. Some conclusions can be summarized as follows:

According to the forecast model described in Eqs. (5) to (7), we adjust the model settings of the input sequence length M, the multivariate coupler coupler(⚫), the attention weights α and β, and the transfer training and then investigate the performance changes. The optimal input sequence length of the model is 3 according to the trade-off between forecast skill and computational resource consumption. It implies that the ENSO deep learning forecast model does not need a relatively long-sequence input. Although the long sequence contains a rich tendency and periodicity of ENSO events, the meteorological chaos is more dominant, which seriously hinders the prediction. Transfer learning is a practical method. Training the model on the reanalysis dataset and subsequently on the remote sensing dataset can effectively reduce the systematic forecast errors by at least 15 %. When replacing the graph-based multivariate air–sea coupler in ENSO forecast model with other deep learning structures, the forecast skill drops obviously. This demonstrates that the graph formalization is a powerful expression for simulating underlying air–sea interactions, and a corresponding the ENSO forecast model with novel multivariate air–sea coupler can forecast more realistic meteorological details. This also demonstrates that it is critical to choose suitable deep learning structures to incorporate prior climate mechanisms for improving forecast skills. The self-supervised attention weights are also promising tools to grasp the contributions of different predictors and memory varieties of different forecast start calendar months. In addition, in comparison with other state-of-the-art ENSO forecast models, the ENSO-ASC achieves at least 5 % improvement in SSIM and PSNR for long-term forecasts.