VIKING20X is an ocean general circulation model configuration with a high-resolution nest in the Atlantic basin. The nest provides eddy-rich resolution not only for the region of interest around the latitude 26.5° N but also at all relevant key regions for AMOC change. The model configuration is based on the NEMO framework and has a global OCRA025 configuration (1/4° resolution) with a higher resolution nest (1/20° resolution) covering the Atlantic from 33.5° S to about 65° N using Adaptive Grid Refinement (AGRIF) . In both, the base and the refined nest component, the configuration has 46 vertical z-levels with a vertical resolution decreasing from 6 m close to the surface to 250 m close to the ocean floor. The model is subject to the JRA55-do atmospheric forcing between 1958 and 2023 . In alignment with the OMIP-II protocol , six cycles of the model simulation each covering the period of 1958 to 2023 were sequentially executed with the same forcing for each cycle. The first cycle was initialized from an ocean at rest and from hydrographic conditions reflecting the World Ocean Atlas 2013 (WOA13, ). Subsequent cycles were started from the last time step of the previous cycle with a restarted forcing period. To account for a spinup time of the model, we only used the first cycle from 1970 onward. The high-resolution simulation VIKING20X is known to resolve mesoscale processes in the entire North Atlantic which includes key processes for the variability of the AMOC such as deep water formation and the process of deep convection observable in the subpolar North Atlantic , the distribution of fresh water in the subpolar gyre , and the overflow pattern , as well as aspects of the wind-driven circulation such as western boundary currents and mesoscale eddies .

The AMOC strength and variability in the first cycle is described in detail in and is used in this study as the reconstruction target. To increase the amount of data that can be utilized for the training process, we also use virtual measurements from the second cycle. The AMOC strength for the first two cycles is shown in Fig. e on a seasonal time scale with 15.5 Sv mean transport and 2.6 Sv standard deviation. While the first of the VIKING20X cycles has a stronger decadal AMOC variability associated with the deepwater formation in the 1990s in the subpolar North Atlantic , the second cycle simulates a reduced decadal AMOC variability . We expect that the generalizability of our trained reconstruction would profit from an increased heterogeneity of the training data as long as the training data and training targets represent consistent physical relations. Hence, even though the weakening decadal variability of the AMOC over the course of the six OMIP cycles may be due to spurious model drift, it may still be an advantage to include data from all six OMIP cycles into our training.

The RAPID Array is a multi-national effort to monitor the overturning at 26.5° N and is deployed since April 2004. RAPID estimates the AMOC in three different contributions, the Ekman transport with remote sensing product , the Florida Current transport via an underwater cable in the Florida Strait , and the upper mid-ocean component. This upper mid-ocean can be divided into the Antilles Current measured by moored current meters and the interior geostrophic transport which is measured by moorings over large distances between the continental margins . We will use the interior geostrophic transport calculation from RAPID in the context of the ocean simulation to investigate how much geostrophic variability can be reconstructed on different time scales with Argo floats.

Argo is an international network of Lagrangian floats which reside and drift with the ocean currents at 1000 m depth and profile the upper 2000 m of the water column in a fixed interval of 10 d . Argo floats provide the methodology to measure the interior of the ocean without any fixed mooring or recurrent ship survey. However, coming with the cost of a sparse and unstructured spatial distribution. Argo floats were first deployed in 1999 and their coverage has continuously grown until 2012 with approximately 430 Argo profiles per season in our study area.

Previous works averaged Argo measurements along larger time frames, creating zonal-depth section of temperature and salinity profiles from which the geostrophic transport is inferred. Also the velocity information of the Argo floats at their parking depth, estimated by the offset of the surface position, was used to estimate transport . Those methods neglect any individual float correlations due to the limitation of unstructured profiles. , who reconstructed the AMOC with bottom pressure used for information about the ocean interior, highlighted the need to investigate how Argo floats can be adopted for this task. To handle the spatial distribution of Argo profiles, we here use a graph data structure , which allows for a flexible topology.

While this study investigates to which extent observation strategies from the RAPID mooring array can be explained by Argo float data, it cannot directly use the observations from the real ocean, because the RAPID array only covers a period of 20 years. The resulting amount of independent samples from 20 years of observational data is not sufficient for training our machine learning framework, on longer than monthly or seasonal timescales. A supervised machine-learning approach is a general function approximation which requires data samples containing an input, the widely available measurements, and a target variable, the interior geostrophic or AMOC transport, to learn the reconstruction. In the following sections and summarized in Fig.  we describe in detail how the input and output is extracted from the ocean simulation.

Argo profiles pose a demanding challenge for the design of neural networks, as neural networks are normally designed to handle structured inputs. For spatio-temporal data points, structured input requires a constant number of measurements from constant locations with a static topology (e.g. a grid structure, or a graph), which is both not true for moving Argo floats. The virtual Argo profiles provide spatial information in form of longitude and latitude, and temporal information in form of a single point in time at which they reach the surface. A static clustering on all data points, could create such a structured input. Around our targeted latitude this would amount to a zonal binning but the set of Argo floats, especially for the shorter target time scales, show a heterogeneous distribution that would require carefully hand-crafted cluster boundaries. A classical binning comes with even more assumptions about shape, distance, and connectivity of the bins, which would require a thorough testing of hyper parameters. However, we aim in this work for a data-driven mapping function which identifies a structured numerical representation (n-dimensional vector) of the underlying ocean state which is independent from specific spatial distribution of the Argo profiles. Often, the term embedding is used for such a n-dimensional vector, but also for the vector space, and for the mapping . Constructing a suitable embedding (see black box in Fig. B) has two main aspects. First, the same ocean state observed by different spatial and temporal distributions of Argo profiles should result in the same (i.e. negligible difference) embeddings. Second, two embeddings should be similar (i.e. small difference) if the corresponding ocean states are similar, too. A successful embedding creates a vector representation of the manifold of ocean states that can be used to generalize for unseen but similar ocean states in the evaluation.

Therefore, we utilize a framework that learns a dynamic spatio-temporal clustering to create structured inputs. The framework Sparse Unstructured Spatio Temporal Reconstruction (SUSTeR, ) was originally developed for road traffic reconstruction from sparse observations in varying locations. SUSTeR set out with the goal to handle unstructured traffic observations and find a general representation of city traffic, much like the unstructured virtual Argo profiles with the goal to find a general representations of the ocean state. For technical details we refer the reader to the original paper, and provide in the following a brief intuition for the oceanographic audience. This framework acts as a learnable clustering with a fixed set of predefined clusters. The assignment decision is non-linear and adds each virtual Argo profile to one cluster based on its spatial and temporal component. A learnable clustering in contrast to the often used static clustering has dynamic boundaries between clusters which can represent different spatial and temporal resolutions, can have non-linear boundaries, or can change with global context (e.g. seasons). After each virtual Argo profile is assigned to one of the Learned Centroids (see Fig. B) the measurements of the virtual Argo profiles are aggregated within each cluster. These clusters represent the structured input, because their boundary will be fixed after training and the amount of clusters is constant. To learn the spatial and temporal correlation between these dynamic clusters the framework SUSTeR uses graph neural networks , which can be thought of as message passing between the clusters, to learn the spatio-temporal correlations. After the clusters exchanged information, all features from the graph nodes (learned clusters) are concatenated and reduced in their dimensionality by a shallow feedforward neural network. The result is not human interpretable but serves as an embedding representing the ocean state.

The output of the Argo processing module is an embedding which is one input of the reconstruction module. We highlight the fact that the mean of all virtual Argo profiles in a single sample is subtracted from the profiles. The mean is separately used in the reconstruction module (see Fig. A), as the interior transport is characterized by a zonal pressure gradient, independent from the mean profile.

Together with the ten zonal wind stress bins, the Florida Current transport, the Antilles Current transport, and the mooring information below the 2000 m Argo depth, the data are concatenated into an input vector for a feed-forward neural network. A feed-forward neural network is a sequence of layers containing a linear mapping and a non-linear activation function. Such an activation layer is the Rectified Linear Unit (ReLU) which is a composite function of the two linear function (constant zero for any negative values and a identity function otherwise). Feed-forward neural networks represent the basic form of neural networks to find correlations in data that do not use information about the hidden neighboring structure of the data, i.e. if the data are to be understood as sequential ordering, as a regular structured image, or as a more flexible topology. Such networks were already used to reconstruct the AMOC in the work of . We found that our architecture requires more neurons compared to that of to handle the complex structure of the Argo processing vector representing the interior ocean state. The network is trained with backpropagation minimizing the mean squared error between the reconstruction and the ground truth interior geostrophic or AMOC transport, respectively. All input and output values are normalized with their respective mean and standard deviation for a faster training .

Further, we discuss the choice for the test, validation, and training periods of our data as it can be seen in Fig. e. While the data in the training period is used to iteratively train the reconstruction, the data in the validation period is used to determine when to stop the training. All evaluations are only based on the data in test period. With the data source VIKING20X we decided to use 2004–2023 from the first OMIP cycle as the test period for the reconstruction because those results compare well with the RAPID time series and other key components of the AMOC in respect to the statistics and temporal evolution . As the task of the validation period is to hinder the leakage of training data into the test period, it should be adequately sized depending on the underlying temporal autocorrelations. We will investigate up to interannual signals, therefore we set aside five-year windows before and after the test period as validation data. While the earlier validation period 1999–2004 is trivial, the second validation period of five years starts with the second forcing cycle. But since the ocean state is continuing into the next cycle, we expect the later validation period to break the auto correlation of the interior ocean state in the same way the earlier period does.

To evaluate the skill of the AMOC reconstruction in comparison to the VIKING20X AMOC, we use four different metrics, the mean absolute error (MAE), the root mean squared error (RMSE), the mean absolute percentage error (MAPE), and the coefficient of determination (R2). For a comprehensive overview we define those skill metrics in the following for a test period with N samples, yi being the targeted AMOC, and y^i being the reconstruction. 1MAE=1N∑i=1N|yi-yi^|2RMSE=1N∑i=1N(yi-yi^)23MAPE=1N∑i=1N|yi-yi^|yi4R2=1-∑i=1N(yi-yi^)2∑i=1N(yi-y‾)2 Large outliers are less visible in the MAE compared to the RMSE where errors are squared. The difference between MAE and RMSE gives an insight whether the reconstruction is able to keep a certain reconstruction quality throughout the test period. Both values are reported in the same unit as the original time series (Sverdrup (1 Sv = 106 m³ s⁻¹)). Because the interior geostrophic and AMOC transport have different amplitudes the relative MAPE allows for an comparison their reconstruction skills. The R2 score reports the amount of variability reconstructed from the target time series with 0 % being a constant mean predictor and 100 % a perfect reconstruction. As we work with neural networks the skill is dependent on the initial random initialization of the weights. In order to produce reliable results we execute each experiment eleven times and report the mean execution skills.

To assess the reconstruction performance across a range of dynamical regimes, we evaluate AMOC reconstructions on timescales from 10 d to one year (Exp. 1 in Table ). Different drivers of the variability play a role on different time scales, which must be investigated to determine on which of the components, the reconstruction performs well and where it shows deficits. Short time scales are known to be driven by the atmospheric forcing in the study area, as shown by , which enabled the AMOC reconstruction from to perform well on monthly data. They found a strong focus of the reconstruction at the wind stress, but they kept the monthly reconstruction and low-pass filtered it to create longer time scales. Here, we show that an already low-pass filtered input will shift the focus onto the density structure, hence increasing importance of the Argo component.

The reconstruction generally captures AMOC mean and variability across all timescales (Fig. ), with decreasing accuracy on shorter time scales due to larger amplitudes. However, the short timescales not only show the largest mean absolute error (Fig. f), also the largest relative error (panel h) with over 10 % mean error on average. We argue that those very short time scales exhibit strong noise, which here we define as variability that is not represented in our input data. Examples for unexplained and therefore not reconstructed fluctuations on the 10 d time scales are changes in the spatial extent of the Antilles Current outside of our bounding box, a sparse amount of Argo profiles that not cover the boundaries or other fine scale processes. For the latter, mesoscale eddies and Rossby waves are potential signals that change the AMOC on such short timescales but are not resolved sufficiently in the input. This shows a limitation of the reconstruction approach for time scales of 10 d or shorter. Despite the limits of the reconstruction on short timescales, we highlight that major features like the weak-AMOC events in end of 2010 or in the beginning of 2013 are well reconstructed . Short-term fluctuations are well captured, although the shortest 10 d time scales demonstrate a slight underestimation at the beginning of the test period (2005–2007) and show reduced variability at a smaller scale in the subsequent years (2019–2021). Most of the skill is lost in the small variations which is the reason for a worse MAE and RMSE compared to a moderate skill decrease in the R2 compared to longer time scales.

Monthly reconstruction (Fig. c) benefits from reduced high-frequency noise and shows a stronger signal-to-noise ratio. Compared to the 10 d timescale, all skill metrics improve for the monthly reconstruction. The signal-to-noise ratio improves for the selected input variables used in the reconstruction, while we achieve an explanation of 74 % of the overall variance (R2 score). It also shows robust performance, as is apparent from the standard deviation of MAE and RMSE across eleven runs with random seeds, which remains below 0.04 Sv.

To interpret the reconstruction process, we apply explainable AI (X-AI) techniques that reveal how each input contributes to the neural network’s estimation of the AMOC. X-AI aims to make typical black-box behavior of neural networks transparent, providing insights into decision-finding process and evaluate the meaningfulness of the inference. The meridional transport at the RAPID array (26.5° N) represents an integrated sum of Ekman transport, transport in the Florida Strait, and upper-mid ocean geostrophic transport . For estimating the AMOC in the real ocean, Ekman transport can be estimated from satellite-based zonal wind stress, and Florida Current is available as a published product. In contrast, the upper-mid ocean transport requires more effort, relying on moorings and repeated ship sections. We hypothesize that Argo profiles have a larger influence on the reconstruction for the longer time series. The main motivation here is that variability beyond the interannual timescales is mainly determined by thermohaline changes which are reflected in the density structure of the open ocean . Experiment 3 (see Table ) calculates the feature importance for each input component (wind stress, Florida Current, Antilles Current, and Argo profiles) and will provide insight in whether the reconstruction acts on longer time scales as expected. Feature importance quantifies how much each input component contributes to the reconstruction. The fully trained network from Experiment 1 (see Table ) defines the baseline performance. By removing the influence of an input component, the degradation of skill scores compared to the baseline describes the feature importance. As we always compare the degradation of the total performance, we achieve comparability among the input components. To remove the influence of one input component, we randomly permute its values in the test data for ten times.

The feature importance analysis reveals how different inputs influence the AMOC reconstruction across time scales. Figure  shows the feature importance with an higher importance being equivalent to a higher increase in the MAE skill when the component is removed from the evaluation. Across all time scales the importance of both currents Florida and Antilles are the lowest and stays constant around a 20 % worse MAE skill score. The black solid lines show the experiments with two-cycles from Fig.  while the dotted black lines represent the six-cycle trained version. Zonal wind stress and the Argo component show the highest importance with up 90 % increase in the MAE. On short time scales the high importance of zonal wind stress is not surprising and agrees with current literature . However, a high importance of wind stress on shorter time scales partially validates the learned AMOC reconstruction, as it utilizes expected and known correlations.

Argo becomes more important on longer time scales, while the wind stress importance stays constant or slightly decreases. For reconstructions trained on the first two OMIP cycles of VIKING20X (black lines), the importance of the Argo component increases up to the seasonal time scales. With both Argo and wind stress importance raising, still the wind stress is more important on time scales up to seasonal for the reconstruction. The turning point is the yearly time scale, which aligns to our hypothesis that on larger time scales, the Argo profiles take the lead from the wind stress for the most important input component. However, the gap between these is marginal for the importance calculated only on two of the OMIP cycles. On the other hand, the feature importance calculated on the six OMIP cycle experiment (Fig. ) shows a more pronounced increasing trend of the Argo profile importance. In summary, reconstructions at longer time scales rely increasingly on the Argo component shown by an increasing focus of the AMOC reconstruction on interior measurements like Argo profiles.

In contrast to Argo profiles and wind stress, boundary currents contribute minimally to the AMOC reconstruction, especially on longer time scales. This finding is in-line with recent work about the signal-to-noise ratio , finding only a small signal in the Florida Current transport. However, feature importance only reflects the importance in a fully trained reconstruction. It does not indicate if an input component can be left out from the input, referred to by fidelity . In VIKING20X, the Florida Current transport time series and the AMOC time series for the 90 d time scale have a linear correlation of 0.58 for the first two cycles shown in Fig. . Removing the influence of the Florida Current transport increases the MAE by only 20 %, suggesting that other inputs carry overlapping information. When calculating the feature importance with regard to the interior geostrophic transport, not shown in this manuscript, we observed a similar pattern, the Florida Current gains importance for seasonal scales, due to the strong correlation to the AMOC . Despite this the importance for all other inputs are constant for the interior geostrophic transport across all time scales with virtual Argo profiles being the most important.

This section investigates whether Argo profiles can help to effectively reconstruct the interior geostrophic transport of the RAPID upper mid-ocean component. The rising importance of Argo profiles at longer time scales (see Fig. ) suggests a stronger role of the geostrophic interior transport in AMOC variability. To validate our reconstruction further in this experiment (see Table  Exp. 4), we define an additional target: a geostrophic interior transport time series (see Sect. ), aligned with the RAPID array's mid-ocean transport component. The contribution of RAPID like interior geostrophic transport we find is to be predominantly negative transport. Between our calculated interior geostrophic transport time series and the AMOC we found a linear correlation of 0.38 on the seasonal timescale showing that the limitation of observations and missing boundary and surface processes change the characteristic of the geostrophic time series. Nonetheless, targeting a reconstruction of this geostrophic time series will provide insight in the suitability of Argo floats to be a proxy for the interior transport as it is measured by arrays like RAPID.

We now evaluate how well the reconstruction framework can capture the geostrophic component of the AMOC. The geostrophic reconstruction uses all input components to utilize all possible data correlations, which is guided by the fact that the different components are not completely independent. For example, it was found by that the Ekman signal contains information about the deeper density structure and the western boundary current. The reconstruction of the interior geostrophic transport (see Fig. ) performs best at seasonal time scales, revealing differences in predictability between AMOC components. On time scales shorter than the seasonal scale, the reconstruction is not comparable to the performance of the AMOC which is expected because the variability of the AMOC is mainly wind-driven . Hence, we not include those shorter time scales in the evaluation.

At seasonal resolution (see Fig. b), the reconstruction captures prominent interannual features of the interior geostrophic transport, suggesting a physically meaningful signal. The time periods with strongly reduced interannual variability in 2013, and 2022 are well reconstructed, suggesting that our reconstruction framework does not compute a static seasonal signal but rather adapts to the dynamical regimes. Although we do not use any absolute timing feature as input (e.g. year, month, or day) from which such a periodical signal could be trivially inferred. However, we believe that information about the phase of the seasonal cycle is still present in the hydrographic data and hence, could still be inferred by the reconstruction. For example, Argo profiles include mixed-layer measurements, which in the subtropical North Atlantic are sufficient to infer the seasonal cycle from surface-near layers. Mostly the seasonal reconstruction struggles with the strong southward events as in 2013, and 2014 by underestimating the amplitude.

Despite improvements at seasonal scales, the reconstruction of the interior geostrophic transport loses accuracy at yearly resolution, pointing to limits in data coverage and training diversity. With the yearly time scale we see a similar drop from seasonal to yearly in the explained variance for the two cycle OMIP training of the AMOC (Fig. g). Interestingly, the reconstruction trained on all OMIP cycles with the geostrophic target variable drops to 50 % of explained variance (Fig. f) which in contrast kept stable for the AMOC. The additional cycles from the two cycle training to the six cycle training in the geostrophic reconstruction increase the performance on the seasonal time scale across all metrics. This improvement was not observed with the AMOC, indicating that more training data benefits the geostrophic time series reconstruction similar to the more training data for the yearly reconstruction of the AMOC. This opens the question whether the yearly reconstruction of the interior geostrophic transport can still be improved by more heterogeneous ocean states in the training data. One option is to reuse the same training samples but using different spatial Argo distributions to augment the existing data; another is adding additional ocean simulations to the reconstructions. However, both approaches begin to address the broader problem of transfer learning for applying reconstructions to real-world data, which we outline in the discussion but lies beyond the scope of this study.

Important for our study is the comparison of the interior geostrophic transport reconstruction with the Argo profiles as described before and a reconstruction trained without any Argo profiles or information about the deep ocean. Such a reconstruction that only utilized zonal wind stress, Florida Current transport, and Antilles Current transport is shown in the gray dashed lines in Fig. a, b and as gray bar in the corresponding skill plots (c)–(f). We retrained the reconstruction in a neural network without argo-related inputs. This removes the influences of the Argo profiles completely from the reconstruction. Across all skill score a strong performance drop is present between the reconstruction with and without Argo profiles, highlighting the importance of interior measurements for our reconstruction. While the seasonal comparison shows that the reconstruction without Argo profiles underestimates the amplitude of the seasonal variability, the reconstruction on the yearly scales can hardly reconstruct any interannual variability but rather predicts a constant mean function. Despite the limited training data for the interior geostrophic transport at yearly time scales the comparison to an reconstruction without any Argo floats pinpoints the importance of the Argo component in our reconstruction.

The standard Argo network, which samples the ocean only down to 2000 m, may not capture key features of deep ocean variability which could influence AMOC reconstructions. Thus, the reconstruction cannot rely on information from layers below 2000 m, raising the question of how essential deeper data is for accurate reconstruction. In the previous experiments, the reconstruction was supplemented with the virtual moorings between the bottom and 2000 m depth. To assess the potential value of deep-ocean information, we compare in this experiment (see Table  Exp. 5) reconstructions using standard Argo profiles, exchanging all Argo profiles with Deep Argo profiles (to 6000 m), and standard Argo profiles supplemented with mooring-derived transport below 2000 m. In the Deep Argo experiment, we assume a hypothetical scenario in which Deep Argo floats offer the same spatial coverage as as the standard Argo array.

Across all timescales, we found that the added value is limited by the seen training data, which did not cover heterogeneous ocean states in the deeper layers, leaving the possible influence of Deep Argo floats undetermined in our training setup. The similar patterns in the reconstruction suggest that information below 2000 m may have limited influence on the reconstruction performance, even though minor differences appear in the skill metrics (MAE, RMSE, and MAPE). The virtual standard Argo profiles combined with the virtual moorings perform best at the 10 d time scale, but show the worst performance at longer time scales. This is even more highlighted in the R2 score in Fig. h, which shows the expected decline in performance at yearly scales. Interestingly, the decay is the strongest for the combined Argo and mooring data.

To understand the performance drop in the mooring-supported reconstruction, we use t-distributed Stochastic Neighbor Embedding (t-SNE ) to assess whether deep ocean states in the test set are represented in the training data. The algorithm is part of the manifold learning family and maps points from a high-dimensional space into lower dimensions visualization purposes. Using t-SNE, we visualize clusters of the virtual moorings (below 2000 m) across the training, validation, as test data. As shown in Fig. i, the test data is almost completely separated from the training data, indicating a significant distribution shift between them.

These findings suggest that while deeper observations can improve reconstructions, their benefit depends on having sufficient representation in the training data. Although not shown in this experiment, we do expect the deeper observation to contain useful information about the stream function at depth. The performance drop occurs because, during the test period, the reconstruction is forced to extrapolate in unseen data regions. To better leverage mooring data, the training must include more heterogeneous deep ocean states that are representative for the test period.