We utilize MAOOAM developed by in our study. MAOOAM consists of a two-layer QG atmospheric component coupled with a QG shallow-water oceanic component. The coupling between these components incorporates wind forcings, radiative and heat exchanges, enabling it to simulate climate variability. MAOOAM has been widely employed in qualitative analyses for various purposes (e.g., ). Moreover, MAOOAM's numerical efficiency allows us to execute numerous climate prediction experiments at a relatively low computational cost.

In MAOOAM, the model variables are represented in terms of spectral modes. Specifically, dax (dox) represents the x-direction resolution, and day (doy) represents the y-direction resolution in the atmosphere (ocean). The model state comprises na (na=day(2dax+1)) modes of the atmospheric streamfunction ψa and temperature anomaly θa, as well as no (no=doydox) modes of the oceanic streamfunction ψo and temperature anomaly θo. Consequently, the model state can be expressed as: 1x=(ψa,1,ψa,2,…,ψa,na,θa,1,θa,2,…,θa,na,ψo,1,ψo,2,…,ψo,no,θo,1,θo,2,…,θo,no) The total number of variables in the model state is 2na + 2no. Note that na is typically larger than no, reflecting the distinct characteristics of the two components in MAOOAM. The atmosphere exhibits faster dynamics and smaller-scale variability, necessitating a greater number of modes to adequately capture its behavior. In contrast, the ocean evolves more slowly and is dominated by larger-scale processes, which can be effectively represented using fewer modes . It is also important to note that variables with lower indices correspond to low-order (large-scale) processes, while variables with higher indices correspond to high-order (small-scale) processes. Like many other models formulated in spectral space, MAOOAM offers flexibility in adjusting the number of atmospheric and oceanic variables by simply modifying the model resolution in spectral space.

In this study, we utilize two different configurations of MAOOAM: one denoted as M56 and the other as M36. The M56 configuration comprises a total of 56 variables, with 20 atmospheric modes (na=20) and 8 oceanic modes (no=8). Specifically, the atmosphere in M56 operates at a 2x-4y (i.e., dax=2 and day=4) resolution, and the ocean operates at a 2x-4y (i.e., dox=2 and doy=4) resolution. On the other hand, the M36 configuration includes 36 variables, with 10 atmospheric modes (na=10) and 8 oceanic modes (no=8), identical to M56. The atmospheric component in M36 operates at a 2x-2y resolution (dax=2, day=2), while the ocean component matches that of M56. Figure  displays time series of three key variables in the true model M56 and the dynamical climate model M36 in spectral space, illustrating their different evolution patterns .

It is important to note that the key distinction between M36 and M56 lies in the atmosphere, where M36 has a reduced number of atmospheric modes, specifically 10 modes fewer than M56 in the y-direction. This difference leads to a lack of higher-order atmospheric modes in M36, thereby unable to capture small-scale variability. The atmospheric error in the y-direction propagates to the atmosphere in the x-direction and the ocean component through the coupling terms in the equations. Consequently, the primary source of model error in this study is attributed to the coarse resolution of the atmospheric component in the y-direction.

The Ensemble Kalman Filter (EnKF) is a flow-dependent and multivariate DA method and has been implemented for climate prediction (e.g., ). The EnKF constructs the background error covariance from the dynamical ensemble. The utilization of an ensemble-based error covariance ensures that the assimilation updates approximately respect to the model dynamics, thereby mitigating assimilation shocks .

All experiments in this study are conducted using the DAPPER package . The overall experimental setup is described in Sect.  and depicted in Fig. . Specifically, we employ the finite-size ensemble Kalman filter (EnKF-N) method proposed by . This method adaptively adjusts the inflation factor, thereby reducing the need for extensive manual tuning and enhancing the performance of the assimilation experiments. It is worth mentioning that we expect no significant alterations in the conclusions of this paper when using the traditional EnKF methods instead of EnKF-N.

We consider the dynamical climate model (described in Sect. ) in the following form: 2xk+1=M(xk), where xk+1 represents the full model state at tk+1, xk represents the full model state at tk and M represents the dynamical climate model integration from time tk to tk+1. The model error at time tk+1 is defined as: 3εk+1=xk+1t-xk+1, where xk+1t represents the true state at time tk+1.

We aim to use ANN to emulate the model error εk+1. Since the truth is not known in practice, the training of ANN uses the analysis increments produced by the EnKF . The architecture of ANN used in this study consists of four layers:

The input layer includes a batch normalization layer , which helps to regularize and normalize the training process.

The error surrogate model can be expressed as follows: 4εk+1′=MANN(xk), where MANN represents the data-driven model built by ANN and εk+1′ represents the model error estimated by ANN. The full state at time tk+1 of the hybrid model can be expressed as follows: 5xk+1h=M(xk)+MANN(xk)

We present the experimental setup in Fig. . The experiments are conducted using two configurations of MAOOAM, as described in Sect. . The configuration with 56 variables (M56, Sect. ) represents the true climate system, while the configuration with 36 variables (M36, Sect. ) represents a dynamical climate prediction system. The experiments depicted in Fig.  are performed as follows:

We integrate the M56 configuration with a time step of approximately 1.6 min for a spin-up period of 30 726.5 years, as specified in . Following the spin-up period, we continue the simulation for an additional 249 years, which we refer to as the “truth”. To generate observations, we perturb the “truth” state using a Gaussian random noise. The standard deviation (σhf) of the noise is set to 10 % of the temporal standard deviation of the true state (xt) after subtracting the one-month running average. Observations are generated every 27 h in spectral space, while the observation operator H is the identity operator (H=I) and is also applied in spectral space.

We split the analysis into two parts:

Training data: The former 124.6 years of the dataset are used to train the ANN parameters to build the hybrid model (Fig. ).

To evaluate the prediction skill of each variable, we employ the correlation and root mean square error (RMSE) skill score (RMSE-SS), which are commonly used metrics in weather forecasting and climate prediction. The correlation is defined as: 6Correlation=∑i=1N(xi-x‾)(yi-y‾)∑i=1N(xi-x‾)2∑i=1N(yi-y‾)2, where x represents the prediction (ensemble mean) and y represents the truth. N is the total number of prediction experiments and is equal to 30 (Sect. ).

The RMSE is calculated as follows: 7RMSE=1N∑i=1N(xi-yi)2, where x represents the prediction (ensemble mean), y represents the truth, and N is the total number of prediction experiments. The RMSE-SS compares the RMSE of the prediction to the RMSE of a persistence prediction. It is defined as: 8RMSE-SS=1-RMSEpredictionRMSEpersistence, where RMSEprediction represents the RMSE between the prediction (ensemble mean) and the truth and RMSEpersistence represents the RMSE between a persistence prediction (where the state remains the same as the initial conditions) and the truth. A positive RMSE-SS indicates that the prediction outperforms persistence and demonstrates skill. On the other hand, a negative RMSE-SS indicates that the prediction performs worse than the persistence and lacks skill. By utilizing the correlation and RMSE-SS, we can assess and compare the skill of the predictions generated by the dynamical climate model and the hybrid model across different variables within the same panel, as shown in Fig. .

To assess the statistical significance of the correlation and RMSE-SS, we perform a two-tailed Student's t-test based on the p-value. For the correlation, the null hypothesis is that the correlation is not significantly different from zero, implying no relationship between the predictions and truth. For RMSE-SS, we perform a hypothesis test to determine whether the squared errors (SE) from the prediction and persistence methods differ significantly. Specifically, we compute the SE for both methods and apply a two-tailed t-test to assess whether their means are significantly different. Assuming sufficiently large sample sizes, the difference between the mean squared error (MSE) can be approximated as normally distributed. This approximation is valid under the conditions that the SE from the two methods are independent and have the same mean value: MSEprediction-MSEpersistence∼N0,sprediction2N+spersistence2N, where sprediction2 and spersistence2 are the sample variances of the SE, and N is the total number of prediction experiments. The resulting p-value represents the probability of observing the given difference (or larger) under the null hypothesis. A p-value below 0.05 is considered statistically significant, indicating that the prediction and persistence methods exhibit meaningfully different error characteristics.

To estimate the uncertainties of correlation and RMSE-SS, we utilize the bootstrap method. We randomly select, with replacement, 30 data points from the 30 prediction experiments and calculate the correlation and RMSE-SS based on this sampled data. This procedure is repeated 10 000 times, resulting in a sample of 10 000 correlation and RMSE-SS values. The standard deviation of this sample is then used to estimate the uncertainties associated with the correlation and RMSE-SS. By conducting the t-test and utilizing the bootstrap method, we can obtain a more comprehensive understanding of the significance and reliability of the correlation and RMSE-SS values obtained from the prediction experiments.

In climate prediction, time-mean quantities such as monthly or annual averages are often used because time averaging reduces the impact of chaotic weather variability, making the underlying climate signals more apparent. They also better meet the practical needs of sectors such as agriculture and energy, where planning is often based on mean conditions. In contrast, predicting higher-order statistics accurately remains challenging due to model limitations and computational costs, particularly when high-resolution Earth system models are required. Nevertheless, there is growing interest in representing complex statistical properties to improve the prediction of extreme events and support climate risk assessments.

To evaluate prediction skill across different time scales, we apply two complementary strategies. For short-term predictions, instantaneous outputs sampled every 27 h are used to represent daily variations. For long-term predictions, model outputs are averaged annually to assess the ability to capture low-frequency variability.

The distinction between short-term (daily) and long-term (yearly-averaged) prediction scales in this study is based on the fundamentally different error growth characteristics of atmospheric and oceanic variables. As illustrated in Fig. , atmospheric variables exhibit rapid error amplification, with a doubling error time of approximately one day and saturation occurring within about ten days. In contrast, oceanic variables demonstrate much slower error growth, with errors roughly doubling over the first year and continuing to grow gradually over the subsequent decade.

Within the coupled model framework, the hybrid model is developed to enhance prediction skill across both short-term and long-term timescales. To evaluate its performance, we adopt 50 d and 60-year prediction horizons as representative benchmarks for the subseasonal-to-seasonal and decadal prediction regimes, respectively. The 50 d prediction reflects the model's capability in capturing fast-evolving atmospheric processes, while the 60-year prediction assesses its capacity to maintain predictability over longer oceanic timescales.

Figure a and c show respectively the correlation and RMSE-SS of the dynamical climate model for both atmospheric temperature θa and streamfunction ψa in the spectral space. We find that the variables in low-order atmospheric modes, such as ψa,2, ψa,3, θa,2 and θa,3, have significant prediction skills over 10 d. While most variables in high-order modes have significant skills within a few days, some do not have prediction skills all the time (i.e. ψa,9, ψa,10 and θa,10). Figure b and d show the correlation and RMSE-SS of the hybrid model for atmospheric variables. For atmospheric temperature, the hybrid model is skillful for up to 50 d for most modes (Fig. b), with a significant reduction in prediction error beyond 10 d for most modes (Fig. d). For atmospheric streamfunction, the hybrid model is skillful in predicting low-order atmospheric modes for up to 50 d and high-order modes for up to 15 d. Overall, the hybrid model has higher correlations and RMSE-SS than the dynamical climate model for atmospheric variables. And the hybrid model exhibits greater improvements in lower-order modes compared to higher-order modes (Fig. a, b, c, and d).

Figure e and g show the correlation and RMSE-SS of the dynamical climate model for oceanic temperature and streamfunction. Since the ocean exhibits slower variability than the atmosphere, we compute annual means for oceanic variables to evaluate the model's prediction skill on interannual timescales. The dynamical climate model demonstrates significant prediction skill for up to 60 years in oceanic temperature across most modes, and in oceanic streamfunction in certain modes. Overall, odd-numbered modes exhibit higher prediction skill than even-numbered modes, related to our experimental design (i.e., the difference in atmospheric y-direction mode resolution between M56 and M36). In addition, the oceanic temperature is more predictable than the oceanic streamfunction in the spectral space. Figure f and h present the prediction skills of the hybrid model. The hybrid model has significant prediction skills in both oceanic temperature and streamfunction in all modes for up to 60 years. It is worth noting that the hybrid model has higher correlations and RMSE-SS than the dynamical climate model, in particular, for oceanic temperature in the first and last modes and oceanic streamfunctions in some modes in which the dynamical climate model has no prediction skill at all (e.g., ψo,2 and ψo,6).

To further demonstrate the advantages of the hybrid model, we use a 10 d lead time for atmospheric variables and a 40-year lead time for oceanic variables as examples to show the prediction skills of the hybrid model in the physical space (Figs. and ).

For atmospheric variables, both atmospheric streamfunction and temperature exhibit similar spatial characteristics (Fig. ). We find that the hybrid model has similar spatial patterns but outperforms the dynamical climate model in most grid points.

For oceanic temperature, the dynamical climate model loses prediction skill over the majority of grid points (Fig. a and c). In contrast, the hybrid model demonstrates significantly higher prediction skill across most grid points (Fig. e and g). For oceanic streamfunction, owing to the slow nature of variability in MAOOAM, the dynamical climate model retains high prediction skill at all grid points even at a 40-year lead time (Fig. b and d). The hybrid model further improves upon this, showing higher correlations and RMSE-SS at all grid points, thereby outperforming the dynamical climate model (Fig. f and h).

For long-term climate prediction, there are additional requirements that the hybrid model must meet. Specifically, the model should be capable of running for extended periods without diverging or exhibiting significant physical instability. In our study, we find that the hybrid model maintains stability and does not experience significant physical instability during the 60-year prediction period.

In summary, the overall performance of the hybrid model surpasses that of the dynamical climate model in both spectral and physical space, demonstrating the advantages of incorporating a data-driven error correction model constructed by ML. This result highlights the potential benefits of leveraging data-driven approaches to improve climate prediction skills.

In this section, we extend our analysis by constructing two additional hybrid models to explore the influence of correcting atmospheric and oceanic errors separately. These models are trained using the same inputs as in the previous section, but are designed to correct either atmospheric errors or oceanic errors. By comparing the prediction skills of the regional averaged variables in physical space among these hybrid models, we gain some insight into the relative importance of atmospheric and oceanic error correction for the overall performance of the climate prediction on different time scales.

In Fig. a, b, e, and f, we present the correlation and RMSE-SS of different models specifically for the atmospheric streamfunction and temperature. We observe that there is minimal difference in prediction skill between correcting only the atmospheric errors (green line) and correcting both the atmospheric and oceanic errors (red line). However, in the early prediction period (less than 20 d), correcting only atmospheric errors has slightly higher skills than correcting both atmospheric and oceanic errors simultaneously. When comparing the hybrid models with the dynamical climate model (blue line), we find that correcting only the oceanic errors (cyan line) does not lead to improvements in atmospheric prediction. It is related to the fact in MAOOAM that the atmosphere mostly drives the ocean but the ocean has too weak influence on the atmosphere for short-term climate prediction .

In Fig. c, d, g and h, we focus on the long-term prediction skill of various hybrid models for the oceanic streamfunction and temperature. Our results reveal that the highest prediction skill over 60 years is achieved when both atmospheric and oceanic errors are corrected (red line). The hybrid models constructed by correcting only atmospheric or oceanic model errors exhibit different performances. For the oceanic streamfunction (Fig. c, g), correcting only oceanic errors (cyan line) does not improve prediction skill. As lead time increases, both the correlation and RMSE-SS metrics indicate a degradation in performance, with skill levels even lower than the dynamical climate model (blue line). In contrast, correcting only atmospheric errors (green line) significantly improves prediction skill within the first 20 years. However, beyond 20–30 years, the skill gradually declines and becomes comparable to that of the dynamical climate model. Notably, the hybrid correction that simultaneously addresses both atmospheric and oceanic model errors (red line) consistently outperforms the dynamical climate model after 30 years, in both correlation and RMSE-SS metrics.

Regarding oceanic temperature (Fig. d and h), correcting only atmospheric errors does not improve the prediction of oceanic temperatures, while only correcting oceanic errors can enhance the prediction skill of oceanic temperatures. Additionally, simultaneously correcting both atmospheric and oceanic errors (red line) can achieve the highest prediction skills at all lead times.

To better illustrate the advantages of the hybrid model, we use one prediction experiment as an example to demonstrate the benefits of correcting model errors for long-term simulations (Fig. ). For atmospheric variables (Fig. a and b), correcting only one component does not effectively simulate the slow frequency atmospheric processes (i.e., low-frequency signals around lead time 20 years), while simultaneously correcting both atmospheric and oceanic model errors (red lines) can better capture this variation. For the oceanic streamfunction (Fig. c), solely correcting oceanic errors (cyan lines) causes a phase change compared to the truth. However, the phase of the other models still matches the truth, with some differences in magnitude and timing. For oceanic temperature (Fig. d), correcting only atmospheric errors leads to the largest deviation from the truth (grey lines) in the first 20 years, which is similar to the dynamical climate model. Correcting the oceanic errors is better, but still poorer than correcting both atmospheric and oceanic errors (red lines), which leads to predictions very close to the truth.

In summary, for short-term atmospheric predictions, correcting atmospheric model errors yields better results, while for long-term simulations, correcting both oceanic and atmospheric errors provides the best predictions.