When applied to gridded climate anomaly fields, EOF analysis transforms spatially distributed variables into an orthogonal basis that maximises explained variance. We use standardized fields because our results show that this preprocessing yields better predictive skill. Let X′ denote the matrix of standardized and latitude-weighted annual near-surface temperature (tas) anomaly fields arranged as a sequence of spatial maps on a 143.144 grid. The EOF decomposition is obtained through eigen analysis of the covariance matrix 1C=cov(X′), which characterises the spatial co-variability structure of the dataset. Diagonalisation of C yields a set of orthogonal eigenvectors (EOF1⟂EOF2⟂…⟂EOFn) and their associated eigenvalues (λ1,λ2,…,λn), where each eigenvalue quantifies the fraction of total variance captured by the corresponding mode. The EOFs therefore define statistically independent spatial patterns ordered by decreasing variance contribution. Projection of X′ onto this orthogonal basis produces the Principal Components (PCs), which represent the temporal evolution of each mode and provide a compact linear description of the dominant variability. In practice, truncation to the leading modes provides a compact and efficient representation of the data, while retaining enough information for prediction.

However, although EOFs are used here as a spatial dimensionality-reduction method, the way this reduced space is constructed can condition the skill of the subsequent prediction model. EOF truncation remains based on variance maximization. Consequently, directions associated with high variance do not necessarily correspond to those that maximise forecast skill or dynamical relevance. This limitation motivates the development of alternative dimensionality-reduction strategies specifically designed to preserve predictability-relevant modes of climate variability.

AEs are neural networks including two components: an encoder, which maps the input data into a lower-dimensional latent representation, and a decoder, which reconstructs the original field from this latent vector. During training, the AE optimizes its internal weights to minimize a reconstruction loss, typically the mean squared error between the input and its reconstruction (Bank et al., 2021). Owing to their ability to capture nonlinear structure, AEs have been successfully applied in numerous domains, including denoising, anomaly detection, data generation, and classification (Gondara, 2016; Vahdat and Kautz, 2021). Their usefulness has also been demonstrated in climate applications, where convolutional AEs have yielded promising results for dimensionality reduction of surface temperature fields (Saenz et al., 2018).

In the present work, the AE is used as a predictability-constrained dimensionality reduction method. Unlike EOFs, which retain modes of maximal variance, the AE is designed to preserve information that is most relevant for prediction. To achieve this, we construct an AE that learns a joint latent representation of the fields at time t and at time t+1. This flexibility is a key advantage of AEs and cannot be reproduced within a purely EOF framework. Aside from this task-specific adaptation, the architecture remains simple and can be viewed as an improved PCA: the encoder and decoder each consist of a single dense layer with linear activations.

The AE is applied to the standardized and latitude-weighted fields X′, reshaped into vectors of dimension (143.144). Let n denote the size of the latent space. The encoder maps the input field at time t to a latent vector: 2LVt=Xt′W1+b1 where W1 and b1 matrix are trainable matrices of size (143.144,n) and (1,n), respectively. The decoder reconstructs both the current and next fields from the latent representation: 3[X^t′,X^t+1′]=AEXt′=decoder(LVt)=Xt′W1+b1*W2+b2 where W2 and b2 map the latent vector back to a vector of dimension 2×(143.144). The latent vector LV thus forms the reduced representation used for prediction.

Training is performed using the Adam optimizer with a learning rate of 0.001. The network is trained for up to 200 epochs with a batch size of 500. Early stopping is applied with a patience of 10 epochs, and the learning rate is reduced by a factor of 0.1 if the validation loss does not improve for five consecutive epochs.

We begin with the LIM, a method widely used to reproduce the dynamics of climate models. The LIM is based on the assumption that the evolution of a system can be decomposed into a linear deterministic component and a stochastic component represented by Gaussian white noise. It corresponds to a multivariate linear Markov process describing the dynamics of the state vector z as: 4dz(t)dt=Bz+ε where B is the linear operator representing the deterministic dynamics, and ε is a Gaussian white-noise term accounting for stochastic variability and assumed to be temporally uncorrelated. The matrix B can be estimated from the lag τ0 and 0 covariance matrices, C(τ0) and C(0), as follows: 5B=τ0-1ln⁡C(τ0)C(0).

The covariance matrix Q of the noise term ε is derived from the fluctuation–dissipation relation (Penland and Matrosova, 1994): 6Q+BC(0)+C(0)BT=0.

The corresponding continuous Markov process is simulated by integrating Eq. (4) with a second-order Runge–Kutta scheme (Penland and Matrosova, 1994).

RC is a recurrent neural architecture designed to emulate nonlinear and chaotic dynamics (Jaeger, 2001; Jaeger and Haas, 2004). Training is limited to the output layer, via ridge regression, making RC computationally efficient and stable compared to traditional RNNs. RC has recently demonstrated strong skill in climate prediction tasks (Nadiga, 2021). We use a standard RC architecture, namely the Echo State Network (ESN). The reservoir state satisfies: 7rt=(1-α)rt-1+αtanh⁡(Wrt-1+Winzt) and predictions follow: 8z^t+1=Woutrt.

To assess the respective contributions of memory and nonlinearity, we consider two simplified variants:

RC_nm, a memoryless RC (α=1, W=0);

Finally, we adapt the classical RC to predict the full field Xt+1′ directly from the reduced representation zt. This modified version is referred to as RCn.

RNNs process sequential data by updating a hidden state that carries information over time (Rumelhart et al., 1986). Their training relies on backpropagation through time (Werbos, 1990), but this procedure is often unstable due to vanishing or exploding gradients (Bengio et al., 1994; Hochreiter and Schmidhuber, 1997; Pascanu et al., 2013).

To mitigate these limitations, more advanced architectures such as the Gated Recurrent Unit (GRU) have been introduced (Cho et al., 2014). GRUs use gating mechanisms that regulate how past information is retained or discarded, enabling the modelling of longer temporal dependencies. The GRU update equations are given by: 9ut=σ(Wuzt+Uuht-1+bu)update gate10rt=σ(Wrzt+Urht-1+br)reset gate11ht̃=tanh⁡(Whzt+Uh(rt⊙ht-1)+bh)candidate hidden state12ht=(1-ut)⊙h̃t+ut⊙ht-1

A GRU-based RNN is used in this study. The network takes as input a sequence of the 15 preceding latent state vectors and is trained to predict the subsequent state. The temporal evolution is learned through a GRU layer with a hidden state dimension of 200. A Dropout layer (20 %) follows to reduce overfitting by randomly deactivating units during training (Srivastava et al., 2014). Finally, a dense layer with linear activation maps the GRU output to the target field.

Each emulator is first tested in a deterministic prediction setting using the past2k simulations (500–2015 CE) from the IPSL-CM6A-LR model (Sect. 4.1). For each emulator, nine ensemble members are used for training and the remaining member for testing. This configuration provides a controlled environment with abundant training data, enabling a systematic comparison and ranking of emulator skill.

To assess generalization, all emulators are evaluated across model experiments from 52 CMIP6-class models (Table A1). Training is performed on each model's available historical ensemble members, and evaluation is carried out on the corresponding piControl simulation. Emulator sensitivity to training-data availability is examined by repeating the training with progressively larger subsets of the historical ensemble members, up to the maximum available for each model.

To enhance emulation skill when training data are scarce, we investigate a multi-model transfer learning strategy. The AE is first pre-trained on all available historical ensemble members from the full CMIP6-model ensemble dataset, and then fine-tuned on the target model. RC-based architectures (e.g., AERCn) cannot directly benefit from transfer learning because the reservoir is non-trainable. To overcome this limitation, we replace RCn with a RNN, enabling both the AE and the RNN predictor to exploit multi-model pre-training. The resulting configurations, referred to as AE_MME and AERNN_MME, are evaluated for their ability to improve emulation skill under limited-data conditions.

Finally, we convert the deterministic emulators into ensemble emulators to enable ensemble forecasts evaluation. The LIM naturally yields ensemble generation through its stochastic component (Sect. 2.2.1). In contrast, AE-based and RC-based emulators do not possess intrinsic stochasticity. We therefore tested several ensemble-generation strategies, including additive Gaussian perturbations applied to the input state, perturbations applied in the latent space, and additive output-space noise based on the covariance of one-year prediction errors. Based on these sensitivity tests, discussed in Sect. 5.3.2, we retained ensemble forecasts generated by perturbing the initial state with additive Gaussian white noise.

Emulator performance is evaluated both on annual timeseries of spatially gridded surface air temperature anomaly fields and on a set of five key climate indices designed to capture large-scale and dynamical aspects of climate variability of each CMIP6 model simulations considered in this study. These indices include annual timeseries of:

the global mean surface temperature (GMST),

More specifically, in order to quantify the degree to which RC adapts to model-specific nonlinear behavior, we use an ENSO nonlinearity indicator denoted α, defined as the quadratic coefficient in the relationship between the first two principal components (PC₁, PC₂) of DJF SST anomalies over the tropical Pacific (10° S–10° N, 80° W–120° E) (Karamperidou et al., 2017): 13PC2=α⋅PC12+b⋅PC1+c. PC₁ characterizes ENSO intensity and closely corresponds to the Niño3.4 index, whereas PC₂ describes the zonal structure of events (Eastern vs. Central Pacific ENSO). Larger values of α indicate stronger ENSO asymmetry and more pronounced nonlinear behavior.

EOFs and AE are first evaluated as dimension-reduction by quantifying the information loss induced by projection into latent space. This is done by comparing a field X′ with its reconstruction from the dimensionality reduction X^′. For EOFs, reconstruction is obtained by projecting the PCs back onto the EOF basis. For the AE, reconstruction is obtained by applying the decoder to the LVs. Their reconstruction performance is assessed in terms of explained variance between the original spatio-temporal fields and the reconstructed ones obtained after reduction to n dimensions. The fraction of variance explained collectively by the n latent dimensions is quantified using the coefficient of determination. Locally, at each grid point, it is defined as: 14R2=1-SSESST where SSE=∑(x′-x′^)2 denotes the sum of squared reconstruction errors and SST=∑(x′-x′‾)2 the total variance of the reference time series x′ at the considered grid point.

The explained variance of the full spatial map is then obtained by aggregating reconstruction errors and total variance over all grid points using: 15Rglobal2=1-∑SSE∑SST.

In addition, beyond the collective contribution of the n-dimensional latent space, the contribution of individual latent dimensions is obtained by reconstructing the field while fixing all other latent coordinates to their temporal mean. The explained variance of this reconstruction, computed using Eq. (14), is the percentage of variance explained by that latent dimension.

Finally, spatial patterns associated with one individual latent dimension are obtained by introducing a spatial reconstruction, in which all latent coordinates are fixed to their temporal mean, expect one which is fixed to its mean plus one standard deviation. By differencing this reconstruction from a baseline in which all latent coordinates are fixed to their mean, allows identifying the spatial pattern associated to a specific dimension of the latent space.

Emulator performance in terms of deterministic prediction at one-year lead-time is then assessed based on two metrics: the Root Mean Square Error (RMSE) and the Pearson correlation coefficient (hereafter correlation or r). RMSE quantifies the average amplitude of the prediction error and is expressed in the physical units of the target variable, while the correlation measures phase agreement between predicted and reference anomalies, independent of any systematic bias or scaling differences. High predictive skill is therefore characterized by low RMSE and high correlation. These metrics are computed both for climate indices and at each grid point, enabling the construction of spatial skill maps that reveal the geographical distribution of emulator performance. Finally, spatial averages of the skill maps are used to obtain domain-mean RMSE and correlation scores, which provide concise measures of overall emulator performance. Spatial averaging can also be restricted to specific regions of interest, such as the tropical Pacific (10° S–10° N, 80° W–120° E), to assess regional predictive skill. Statistical uncertainty in correlation and RMSE estimates is assessed using a 10-year block-bootstrap procedure. We generate 1000 pseudo-series by resampling 10-year blocks with replacement from the original time series. For each pseudo-series, correlation and RMSE are recomputed, and 95 % confidence intervals are defined as the 2.5th and 97.5th percentiles of the resulting bootstrap distribution.

To contextualize emulator performance, we also include a baseline benchmark based on Persistence. The Persistence experiment is a conservative benchmark used to evaluate predictive skill. It consists of predicting for the future the same value as the present, and allows quantification of the added value of the emulator-based forecast (Jebri and Khodri, 2023).

To disentangle the respective contributions of memory and nonlinearity within RC prediction skill, we compare the full model skill with two simplified variants: a linear version (RC_ℓ) and a memoryless version (RC_nm). Their relative contributions are quantified as: 16RC% nonlinearity=(RC-RCl)RCl,RC% memory=(RC-RCnm)RCnm.

Comparing these metrics with α provides insight into how effectively RC captures the nonlinear and dynamical characteristics of each climate model.

For ensemble predictions, we use the Continuous Ranked Probability Score (CRPS), a metric widely used to evaluate probabilistic forecasts (Hersbach, 2000; Gneiting and Raftery, 2007; Berrocal et al., 2007). The CRPS measures the difference between the predicted cumulative distribution function (CDF) from the ensemble and the observed value, generalizing the mean absolute error to probabilistic forecasts. Lower CRPS values indicate better probabilistic skill and better-calibrated ensembles. Let F denote the predictive CDF of the ensemble for a given climate index, and let I denote the corresponding observed index value. The CRPS is defined as: 17CRPS(F,I)=∫-∞∞(F(z)-1{z≥I})2dz.

As discussed in Sect. 5.1, emulator skills involve a trade-off between predictive capability and dimensionality reduction. As shown on Fig. 2, the 1-year lead-time prediction skill increases rapidly with latent-space dimension before reaching a saturation regime. For the global mean of the local scores, the GMST, the Nino3.4 and NH climate indices, all emulators reach near-asymptotic correlation values with approximately 9 latent dimensions (Fig. 2a–d). For the Southern Hemisphere mean temperature, the AE and AERCn emulators begin to level off at around 11 latent dimensions, whereas the LIM stabilizes only at higher dimensions (around 19) with a correlation skill only slightly higher than Persistence (Fig. 2e). Finally, for the AMV index, emulator skill saturates much more gradually, with a plateau reached only at relatively large latent dimensions, around 20–25 (Fig. 2f). Consistently across all considered metrics, at least one emulator outperforms the Persistence benchmark once the latent space reaches a dimension of 10. To balance predictive skill with the need to reduce the size of the problem, a latent dimension of 16 is selected for the remainder of the study, as it provides a good compromise across all considered climate metrics with correlation values of at least 0.5. This choice ensures robust forecast skill while retaining approximately 60 % of the total variance (Fig. A6). Compared to a 30-dimensional representation, this corresponds to a reduction of only about 10 % in explained variance, while maintaining similar predictive performance in a latent space that is nearly twice as compact. Repeating the diagnostics presented in Sect. 5.1 using a 16-dimensional latent space yields qualitatively similar results with a subset of AE LVs highly correlated (>0.7) with PC₁, albeit for weaker correlations with the second EOF (Figs. A7 and A8).

With the latent-space dimension fixed, Fig. 3 shows the local skill in predicting the surface temperatures simulated by the IPSL-CM6A-LR model at a lead-time of 1 year for all emulators and for the Persistence benchmark. Table 2 summarizes the corresponding skill metrics averaged over the globe, over key regions and for specific indices. The corresponding confidence intervals at 95 % of correlation scores, estimated using the block-bootstrap procedure (Sect. 4.3.3), are reported in Table A2. Results are discussed by progressively increasing model complexity along the three axes of improvement identified above, starting from Persistence and the LIM, and then considering RC, RCn, AE, and finally AERCn.

Persistence generally exhibits low overall skill, yet it can match or even exceed LIM performance in specific regions. As shown on Fig. 3a–d, Persistence outperforms LIM over the North Atlantic, a result consistent with the particularly strong autocorrelation of surface air temperature in this region in the IPSL-CM6A-LR model (Fig. A9). LIM reveals pronounced spatial disparities in the predictive skill. It performs particularly well in the tropics, especially over the western Pacific, where correlation scores exceed 0.7 (Fig. 3c; Table 2). The observed zonal asymmetry in skill across the tropical Pacific may be explained by the fact that variability in the eastern Pacific is more nonlinear (Sun et al., 2016; Zhao and Sun, 2022), which can limit the effectiveness of linear models in capturing complex dynamics. Moreover, predictive skill is generally higher over the oceans than over land, reflecting the longer memory and greater persistence of oceanic processes, whereas variability over continents is more strongly influenced by short-lived atmospheric fluctuations (Boer, 2000; Bellucci et al., 2015).

Replacing LIM with a RC model leads to consistent improvements across all performance metrics (Table 2). Unlike LIM, RC outperforms Persistence for the NH index. Figure 3e and f illustrates a general, albeit moderate increase in the spatial distribution of predictive skill, indicating that RC's memory and nonlinear processing capabilities contribute to enhanced forecast performance. Along the second axis of improvement, the architectural simplification introduced in RCn yields additional gains, with skill levels clearly exceeding those of LIM (Table 2). RCn outperforms both LIM and Persistence over the North Atlantic and, more notably, over the Southern Hemisphere (Fig. 3g and h), where correlation scores exceed 0.6 (Table 2). Further gains are also apparent in the tropics, particularly in the correlation patterns (Fig. 3g).

The first two axes of improvement already lead to substantial progress in emulation skill. The third development concerns dimensionality reduction. Using the AE alone for both dimensionality reduction and prediction results in further enhancements in model effectiveness. In particular, a clear increase in Niño 3.4 correlation and a marked reduction in Niño 3.4 RMSE are observed (Table 2). Spatially, better scores are especially pronounced over central and eastern tropical Pacific and over Southern Ocean (Fig. 3i and j). Although the AE used here is linear, it nevertheless outperforms the RCn configuration, indicating that the choice of dimensionality reduction method can have a stronger impact on emulation performance than the choice of the prediction algorithm itself.

Finally, combining all three axes of improvement into the AERCn configuration yields the best overall emulator. AERCn consistently outperforms LIM, Persistence, and all other emulators across all metrics and regions (Table 2). The global-mean of local correlation increases from 0.493 for LIM to 0.557 for AERCn. For the Niño 3.4 index, correlation improves from 0.593 to 0.724, and for the SH index, from 0.658 to 0.791, substantially exceeding Persistence (0.704). Spatially, many regions that previously exhibited only moderate skill with the LIM (Fig. 3c) now exceed correlation values of 0.5, as illustrated by the expanded yellow and orange areas in Fig. 3k. Overall, these results demonstrate that progressively increasing model complexity along the three identified axes leads to systematic and robust improvements in emulation skill, both spatially and across key climate indices, with AERCn providing the best overall performance. This hierarchy is supported by the 95 % confidence intervals reported in Table A2, which confirm the robustness of the main skill differences when accounting for uncertainty.

Although global correlation and RMSE scores indicate strong overall performance, a closer examination reveals substantial temporal variability in emulation skill. The running RMSE of the 1-year lead-time predictions obtained by the LIM and AERCn emulators, for the GMST and Niño 3.4 indices exhibits notable fluctuations revealing distinct phases of persistent increased or decreased prediction error at decadal to secural timescale (Fig. 4a and b). In particular, increases in the RMSE of the prediction for the GMST index mostly coincide with peaks in stratospheric volcanic aerosol concentration as depicted by the global mean stratospheric aerosol optical depth, suggesting that these variations are largely attributable to episodes of major volcanic eruptions (Fig. 4a and b). Consistently, the past2k simulations including variations of the Solar Irandiance only (past2k-SSI), display a significant reduction in the amplitude of GMST multi-decadal to secular prediction skills variations (Fig. A10a). In contrast, for the Niño 3.4 index, temporal fluctuations in RMSE do not appear to be related to volcanic activity (Fig. 4b), as similar multi-decadal prediction skill variations are observed even in the absence of eruptions (Fig. A10b). These fluctuations are instead primarily attributable to the intrinsic changes in the amplitude of ENSO events over time as shown by the significant correlation between, the 30-year running RMSE of the Niño 3.4 index prediction and the running standard deviation of the Niño 3.4 index of the past2k simulation itself (Fig. 4c). This relationship is particularly strong for the LIM (correlation of 0.67), which tends to underestimate or fails to predict extreme ENSO events and thus explains less variance during periods of intense ENSO activity. By contrast, this relationship is much less pronounced for the AERCn (correlation of 0.38), illustrating overall better skills at predicting the full range of amplitude of ENSO variability.

The results from the previous section, based on simulations of the IPSL-CM6A-LR over the CE, demonstrate the ability of our emulators to well capture the dynamics of a CMIP6-class model. In this section, we generalized this approach to a full set of CMIP6 models (Table A1) under the evaluation protocol defined in Sects. 4.2 and 4.3.

Each climate model is governed by distinct physical parameterizations, and exhibits specific biases and dynamical behaviors (Zhang et al., 2023; Richter and Tokinaga, 2020; Zhu et al., 2020). For instance, the IPSL-CM6A-LR model is known to exhibit a strong multidecadal variability in the North Atlantic (Boucher et al., 2020). Moreover, combining simulations from multiple models often yields performance metrics that better align with observed climate variability, due to a partial compensation of individual model errors and structural uncertainties (Ahmed et al., 2019; Parsons et al., 2021) stressing the importance of emulating the full diversity of physical processes represented across the model ensemble. Based on these considerations, the comparative analysis across CMIP6-class models is guided by several key questions:

Can RC-based emulators adapt to models with varying degrees of nonlinearity?

Section 5.2.1 showed that replacing LIM with RC improves emulator performance, suggesting a beneficial role for RC's memory and nonlinear processing. Since the IPSL model is relatively linear, it is not clear whether these gains arise mainly from memory or from nonlinear effects. We expect the balance between the two to vary with the degree of nonlinearity in the considered model emulated. This raises the following question: can the RC adapt to the characteristics of each model, selectively leveraging its memory and nonlinear features when needed? To investigate this, we rely on the RC variants introduced in Sect. 2.2.2, and follow the experimental protocol described in Sect. 4.2.1.

For this first experiment, we only consider 16 CMIP6 models holding at least 10 historical ensemble members in order to ensure sufficient training data for RC. The impact of training set size on emulator performance will be explicitly assessed in a subsequent analysis. Moreover, because we focus on nonlinear dynamics in the tropical Pacific, as quantified by the nonlinearity coefficient α (Sect. 4.3.3), we first evaluate RC skill in this region. Figure A11 shows that the spatially averaged correlation exceeds 0.6 between the obtained 1-year lead time prediction and the piControl target over the tropical Pacific for most of the 16 CMIP6 models including IPSL-CM6A-LR, consistent with the results obtained with the past2k simulations (Table 2). We then proceed to a skill attribution analysis with the percentage of skill gain attributed to RC dynamical memory and nonlinear processing over the tropical Pacific region (Fig. 5a). These contributions vary substantially across models. In the case of IPSL-CM6A-LR, approximately 10 % of the RC skill can be attributed to memory effects. By contrast, for models such as CESM2, nonlinear processing contributes more to predictive skill than memory. On average across models, memory accounts for a larger fraction of the skill gain (around 5 %) than nonlinearities (around 3 %). Figure 5b presents a linear regression between the fraction of skill gain attributed to RC memory and the coefficient α, used here as a proxy for model nonlinearity associated with ENSO. The contribution of memory decreases with increasing α, indicating an inverse relationship between memory-driven skill and the degree of nonlinearity in the tropical Pacific. In other words, the more linear the model, the larger the contribution of memory to RC performance. This result confirms that RC adapts to the dynamical characteristics of the model it emulates, relying preferentially on dynamical memory or nonlinear processes depending on the degree of linearity of the underlying system.

As previously noted, this analysis was restricted to models with at least 10 historical ensemble members to ensure sufficient training data. This naturally raises the second question: to what extent does the size of the training set influence predictive skill, and is the hierarchy of emulators performance observed with the IPSL-CM6A-LR model preserved across the full set of CMIP6-class models?

In this second experiment, we extend the analysis to all 52 CMIP6 models, training the emulator with progressively increasing subsets of historical ensemble members, following the protocol of Sect. 4.2.1. Figure 6 shows the 1-year lead-time correlation between the surface temperature prediction and the target as a function of the length of the training period across the full set of CMIP6 models. The LIM reaches its optimal performance relatively quickly regardless of the model (Fig. 6a and b), requiring only 495 years of training data (equivalent to 3 historical members). In contrast, for all RC-based methods, predictive skill strongly depends on the size of the training set (Fig. 6a, c, d, and f). On average, 1320 years of training (about 8 members) are needed for RC-based emulators to perform as well as the LIM. Similarly, the AE skill stabilizes around 1320 years, but it requires only about 3 members (495 years) to reach performance levels comparable to the LIM (Fig. 6a and e). Overall, the best-performing emulator depends on the amount of training data available: the LIM performs best when the training set is very limited (fewer than 495 years), the AE is superior for intermediate training sizes (up to approximately 2310 years), and the AERCn becomes the most effective option when training data are abundant (Fig. 6a). In this last case, the hierarchy among emulators is consistent with the results presented in Sect. 5.2.1 (Fig. 3; Table 2).

However, these results also reveal the limitations of the emulators in outperforming the LIM when training data are scarce. To address this issue, we investigate a transfer learning approach designed to mitigate the impact of limited training sets, following the experimental protocol described in Sect. 4.2.2. Since RC cannot be used in a CMIP-multi-model ensemble (MME) setting due to its non-trainable nature, we replace it with an RNN (Sect. 2.2.3), enabling the construction of a CMIP6-multi-model equivalent of AERCn, referred to as AERNN. This results in two multi-model emulators, denoted AE_MME and AERNN_MME. They are first pre-trained using one historical ensemble member from each of the 52 CMIP6-class models. This pre-training phase enables the emulator to learn a shared physical basis across a wide range of climates. The models are then fine-tuned individually for each target model, starting from the pre-trained weights.

Figure 7 highlights the clear advantage of multi-model and transfer learning approaches under data-limited conditions. It shows the correlation of the 1-year lead-time prediction with the target as a function of the length of the training period across the full set of CMIP6 models. The AE_MME and AERNN_MME curves start at significantly higher skill levels compared to their their single-model counterparts (Fig. 7a) regardless of the CMIP6 host-model (Fig. 7c–f). Overall, the AERNN_MME emerges as the best-performing emulator when training data are very limited (<660 years; Fig. A12). However, when training data are abundant, it is noteworthy that the multi-model version offers no advantage over the classic AERNN (Figs. 7a and A12).

Figure 8 shows, for the LIM and the AERCn, the RMSE of the one-year lead forecast of surface air temperatures of the IPSL-CM6A-LR and MIROC6 piControl simulations, computed exclusively during extreme El Niño years and Extreme La Niña years. Extreme warm and cold events are defined as Niño 3.4 anomalies exceeding one standard deviation (Santoso et al., 2017). For IPSL-CM6A-LR, the LIM exhibits relatively large errors in the eastern tropical Pacific during extreme events (Panels a and c), particularly for extreme La Niña years. AERCn substantially reduces the overall large-scale errors (Panels b and d), including the tropical Pacific although this region remains the most challenging. With MIROC6, LIM errors are globally comparatively smaller and in particular over the tropics (Panels e and g). Over the western Pacific, the skill during extreme El Niño years is consistent with the stronger persistence simulated in this region by the MIROC6 model (Fig. A14). AERCn nevertheless yields a systematic reduction in RMSE broadly and over the eastern tropical Pacific (Panels f and h), indicating a robust gain in skill under both extreme ENSO phases.

To further characterise emulator performance at forecasting extreme ENSO events, we focus specifically on the Niño 3.4 index and categorize Niño 3.4 anomalies into three types: extreme El Niño, extreme La Niña, and Moderate events (defined as Niño 3.4 anomalies below ±1 standard deviation). Figure 9 presents the RMSE of the 1-year lead-time prediction obtained by the LIM, RC, AE, and AERCn, stratified over the three ENSO event types. As expected, prediction errors are larger for extreme events and in both cases, replacing the LIM with the AERCn emulator substantially reduces RMSE, yielding an average improvement of approximately 20 % relative to the LIM. However, a notable contrast emerges between the two CMIP6-class models: errors are largest during extreme La Niña events for IPSL-CM6A-LR, whereas they peak during extreme El Niño episodes for MIROC6 (Fig. 9). This indicates a general asymmetry in skill between warm and cold events that depends on the host-model characteristics. In particular, these asymmetries reflect the distribution of extremes in each data-set: IPSL-CM6A-LR tends to produce stronger La Niña than El Niño events, leading to higher RMSE values when emulators attempt to reproduce these large-amplitude cold events. MIROC6 exhibits the opposite pattern. This result is consistent with broader findings for CMIP6-class models. Many models simulate ENSO dynamics that are too linear, often generating La Niña events that are excessively strong compared with observations (Zhao and Sun, 2022). Only a limited subset of CMIP6 models, including MIROC6, realistically captures ENSO asymmetry. The more linear ENSO dynamics of IPSL-CM6A-LR may therefore lead to an under-representation of ENSO asymmetry and to amplified cold anomalies, which in turn likely increases emulator forecast errors during extreme La Niña events.

Accurately predicting the evolution of the climate system requires not only a reliable representation of its underlying dynamics, but also a robust quantification of uncertainties. Uncertainties arise from imperfect knowledge of initial conditions, unresolved nonlinear processes, and external forcing. In this context, confidence in purely deterministic forecasts is limited. Relying solely on such forecasts may produce an overly narrow and potentially misleading representation of the range of possible future climate states.

To overcome this limitation, probabilistic approaches based on ensemble forecasting are required. Monte Carlo methods, such as Particle Filters, approximate the probability density of the climate system using an ensemble of particles (Jebri and Khodri, 2023). The predictive components and the associated ensemble spread are essential for representing the range of possible future states and for quantifying forecast uncertainty. The LIM framework naturally supports ensemble forecasting through the stochastic term included in its governing equations (Sect. 2.2.1), allowing a direct transition from deterministic to probabilistic predictions. By contrast, AERCn does not include an intrinsic stochastic component. Figure A15 presents a sensitivity test of AERCn ensemble generation for IPSL-CM6A-LR, based on three perturbation strategies. The upper row shows GMST rank histograms (Fig. A15a–c), and the lower row shows Niño3.4 rank histograms (Fig. A15d–f). The columns correspond to input-field perturbations (Fig. A15a and d), latent-space perturbations (Fig. A15b and e), and output-space stochastic noise based on the covariance of one-year prediction errors (Fig. A15c and f). CRPS values are indicated in each panel. A flat rank histogram indicates a well-calibrated ensemble, meaning that the target trajectory is statistically indistinguishable from any ensemble member and behaves like one random realization of the system. Latent-space perturbations produce results close to input-space perturbations for Niño3.4, but lead to a more peaked GMST rank histogram, suggesting over-dispersion. Output-space noise performs less well for Niño3.4, with a U-shaped rank histogram indicative of under-dispersion and substantially higher CRPS values. Overall, input-space perturbations provide the best compromise across both indices. We therefore generate AERCn ensembles by perturbing the initial conditions with spatially uncorrelated white noise, with an amplitude set to 2.5 times the grid point standard deviation of the anomalies.

Using this perturbation strategy for AERCn, we then evaluate the ability of the LIM and AERCn models to capture the plausible range of outcomes of the emulated system and including extreme events. Figure 10 presents rank histograms computed from 100-member ensemble forecasts at a 1-year lead time for the GMST and the Niño 3.4 index, using both LIM and AERCn on the IPSL-CM6A-LR and MIROC6 models. Overall, both LIM and AERCn produce well-calibrated ensembles for the GMST and Niño 3.4 indices. The rank histograms are generally flat, and the target lies within the ensemble spread 97 %–99 % of the time (Fig. 10a–h). MIROC6-based rank histograms show a slight concentration in the central ranks for the GMST index (Fig. 10c and d), suggesting a tendency toward over-dispersion, whereas the Niño-3.4 histograms display the opposite pattern, with somewhat elevated frequencies in the non-central in-spread bins, particularly for the LIM emulator (Fig. 10e and g).

While rank histograms provide valuable insight into ensemble shape and calibration, they are insufficient for a comprehensive assessment of probabilistic forecast quality. To obtain a quantitative and statistically robust measure of ensemble skill, we therefore use the Continuous Ranked Probability Score (CRPS), a proper scoring rule that jointly evaluates forecast accuracy and calibration (Sect. 4.3.4). Figure 11 displays the distribution of CRPS values computed during extreme ENSO years, defined as years in which the Niño 3.4 index exceeds ±1 standard deviation. For each extreme year, CRPS is calculated for the Niño-3.4 index (panels a and b) and for GMST (panels c and d). Results are shown separately for IPSL-CM6A-LR (a, c) and MIROC6 (b, d). Each violin represents the full distribution of CRPS values across extreme-event years. The width of the violin indicates the density of values: broader sections correspond to more frequent CRPS values, while narrow upper tails indicate rare but large forecast errors. Accordingly, AERCn consistently yields lower CRPS values than the LIM across all cases (including the upper tails), indicating superior probabilistic skill. This result demonstrates that AERCn produces more reliable and better-calibrated ensemble forecasts during extreme ENSO conditions, in agreement with the deterministic performance gains reported in Sect. 5.3.1.