Introduction

The Antarctic climate system is characterized by complex interactions among the atmosphere, ocean, and cryosphere, resulting in a region of high climate sensitivity and pronounced variability1. As a fundamental component of this system, the Southern Ocean (SO) plays a pivotal role in modulating global weather patterns through its influence on oceanic processes within the Earth’s climate system2. Through its complex interactions with atmospheric circulation3 sea ice dynamics, and deep-water formation, the SO plays a crucial role in shaping large-scale patterns of heat distribution, carbon cycling, and climate variability4. Within this context, ocean surface waves in the SO represent a complex and vital parameter, significantly affecting air-sea interactions5,6 including energy transfer7 and coastal dynamics8. The SO is characterized by extreme weather conditions and complex atmosphere-ocean coupling, making the accurate prediction of significant wave height (SWH) essential for operational forecasting and advancing climate research9. SWH, a key metric influencing coastal erosion, ice-sheet dynamics, and navigational safety in Polar Regions, with its variability driven by atmospheric forcing, ocean currents, and sea ice conditions. Therefore, a detailed understanding of ocean surface wave behavior is crucial to elucidate the SO’s role in the global climate system.

Generally, numerical models based on physical principles—such as WaveWatch III (WWIII)10 and SWAN (Simulating Waves Nearshore)11—have widely used for SWH prediction12,13. These models simulate wave generation, propagation, and dissipation by solving partial differential equations and rely heavily on input fields like wind forcing, bathymetry, and boundary conditions. However, uncertainties in these inputs—due to limited observational coverage, spatial and temporal mismatches, and measurement errors—can propagate through the models, significantly degrading forecast accuracy. Furthermore, the nonlinear dynamics of waves, especially in data-sparse and high-latitude environments like the SO, compound these challenges. Consequently, deterministic wave models often face limitations in generalizing effectively and providing reliable forecasts under highly variable or extreme conditions.

Recent advances in machine learning (ML) have shown promise in addressing these challenges. Studies demonstrate that long short-term memory (LSTM) networks and hybrid models incorporating principal component analysis (PCA) outperform traditional regression and support vector machine (SVM) methods in forecasting polar wave heights and periods, particularly in sparse data environments14. Deep learning frameworks employing convolutional neural networks (CNNs) and ConvLSTM architectures have further improved SWH predictions by explicitly modeling wind-wave energy transfer processes, achieving superior performance relative to conventional spectral models in capturing nonlinear interactions15. Transformer-based architectures trained on multi-decadal reanalysis data have reduced prediction errors of extreme waves by 20–30% compared to classical numerical models16. Additionally, hybrid CNN-LSTM models have enhanced SWH retrieval from SAR altimetry by resolving nonlinear wave evolution patterns, while explainable ML techniques such as SHAP-enhanced gradient boosting provide new insights into the polar vortex dynamics and precursors to extreme events17,18. Convolutional-recurrent neural networks leveraging reanalysis datasets (e.g., ERA5) have also shown effectiveness in spatiotemporal wave height forecasting by simultaneously capturing geographic and temporal variability19. Physics-informed neural networks that incorporate wave dynamics constraints (e.g., spectral energy conservation) have improved accuracy for extreme event prediction in data-sparse regions while maintaining physical interpretability, further illustrating ML’s transformative potential for polar wave forecasting20. Accurate and timely wind forecasting21,22 is crucial for improving wave prediction, as wind is the primary driver of wave generation in the oceans.

Despite these advances, delivering accurate and reliable wave height predictions under the extreme, nonlinear, and data-sparse conditions typical of the SO remains an unresolved operational challenge. Usual numerical models like WWIII and SWAN suffer from compounded uncertainties in wind forcing, bathymetry, and sea ice that amplify forecast errors in polar environments. While ML approaches—including artificial neural networks (ANNs)23 SVMs24 and random forests25—have been applied predominantly to coastal and midlatitude regions, they rarely address extrapolation to unseen extremes or the unique complexities of polar zones. This study addresses this gap by developing an Adaptive Neuro-Fuzzy Inference System (ANFIS) model26 optimized with subtractive clustering and rigorously validated against independent, out-of-sample SO wave data. This approach meets a critical scientific and operational need for robust forecasting in remote, high-risk maritime environments. Key parameters driving SWH prediction include surface wind speed (SWS), MSLP, wave period, and water depth, with SWS recognized as a primary forcing variable15. ML and ANFIS models leverage these meteorological inputs to enhance forecasting accuracy27. Nonetheless, challenges remain, especially concerning the availability of high-quality, long-term observational data and the interpretability of complex ML models in extreme and data-limited environments such as the SO.

To better understand the influence of atmospheric variables on SWH, we employed a feature correlation heatmap analyzing relationships between SWH and parameters including dewpoint temperature (DPT), 2-meter air temperature (2mT), MSLP, sea surface temperature (SST), surface pressure (SP), and wind speed. This analysis indicates that wind speed and SP exhibit the strongest correlations with SWH (Fig. 1).

Fig. 1
figure 1

Feature correlation heatmap. Relationships between SWH and meteorological variables: dewpoint temperature [DPT], 2-meter temperature [2mT], mean sea level pressure [MSLP], sea surface temperature [SST], surface pressure [SP], surface wind speed [wind].

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Despite recent advancements in machine learning for SWH forecasting, significant challenges persist in achieving accurate wave predictions in polar environments like the Southern Ocean. Traditional numerical models, such as WWIII and SWAN, often suffer from reduced accuracy due to uncertainties in input data (e.g., wind forcing and bathymetry) and the complex nonlinear dynamics of waves in data-sparse regions. Existing machine learning approaches, including artificial neural networks, support vector machines, and random forests, have primarily been developed for coastal and mid-latitude regions, exhibiting limitations in generalizing to extreme or unseen conditions in polar zones. This study addresses this research gap by introducing an ANFIS optimized with subtractive clustering. The proposed model leverages sequential time-lagged inputs and automated fuzzy rule generation to effectively capture nonlinear wave dynamics, enhancing generalization to unseen data in polar environments. By focusing on key variables such as surface wind speed (SWS) and mean sea level pressure (MSLP), this approach not only improves prediction accuracy but also provides an interpretable tool for operational forecasting in high-risk maritime regions.

Methodology

Study area

The SO serves is a critical component of Earth’s climate system, acting as both a major carbon sink28 and driving of global thermohaline circulation via the Antarctic Circumpolar Current (ACC)29. This unique current system enables inter-basin exchange between the Atlantic, Pacific, and Indian Oceans, while sustaining distinct marine ecosystems that are sensitive indicators of climate variability30. Accurate quantification and prediction of SWH in the SO are vital for numerous applications including maritime safety, search-and-rescue operations and navigation optimization. Moreover, variability in SWH provides key insights into air-sea interaction dynamics31 and coastal geomorphological processes32. Enhancing SWH forecasting capabilities will therefore contribute substantially to operational safety and to advancing our understanding of climate-related ocean processes. The spatial distribution of observational platforms employed in this study is illustrated in Fig. 2.

Fig. 2
figure 2

Study area showing buoy positions in the SO. The map was generated using MATLAB R2023a (Mapping Toolbox).

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Data collection and preprocessing

The SO is a vital component of the Earth’s climate system owing to its significant role in carbon dioxide absorption and in driving global ocean circulation via the ACC. In collaboration with Spoondrift and the Defence Technology Agency, MetOcean Solutions deployed six wave buoys—including five drifting Spotters—to collect real-time wave data in this remote region [www.metocean.co.nz/southern-ocean]. To integrate buoy observations with ERA5 reanalysis data, ERA5 variables were extracted for each buoy’s latitude and longitude by applying nearest-neighbor interpolation on the 0.25° × 0.25° ERA5 spatial grid. Both buoy and ERA5 time series were then aligned to uniform 3-hourly timestamps. When ERA5 data points did not perfectly coincide with buoy measurements, linear temporal interpolation was employed to fill gaps of up to one time step (3 h). This approach ensured synchronization of in situ (buoy) and reanalysis (ERA5) variables within each recorded observation used as input for modeling. The buoy systems are solar-powered, easily deployable, and engineered to withstand extreme environmental conditions, thus providing high-quality data on wave heights and associated climate interactions.This study analyzes MetOcean buoy and ERA5 data spanning 2019 to 2020, encompassing SWH, DPT, 2mT, MSLP, SST, SP, and surface wind speed. A preliminary feature correlation heatmap (Fig. 1) was employed to identify the meteorological parameters most relevant to SWH, resulting in the selection of MSLP, SWS, and SWH for subsequent predictive modeling. Data were recorded at consistent 3-hourly intervals, providing a comprehensive temporal coverage of oceanic and atmospheric conditions.

To ensure data integrity before model development, a thorough cleaning procedure was applied. The dataset was screened for missing, duplicated, or anomalous entries using MATLAB’s data preprocessing tools. Although missing values were minimal due to the high-quality nature of the buoy and ERA5 datasets, any incomplete records were removed to prevent bias. Erroneous data—such as wave heights exceeding physically plausible bounds or negative pressure values—were excluded. After this validation and cleaning, 2,673 samples remained for modeling. For temporal consistency and rigorous evaluation, the dataset was chronologically partitioned into three subsets: 70% for training (to develop the model), 30% for testing (to evaluate generalization within observed data distributions), and an additional distinct set of 250 subsequent records reserved exclusively for extrapolation analysis. This approach simulates real-world forecasting conditions by assessing model performance on unseen future data.

Model architecture

To effectively capture the nonlinear relationships between meteorological variables and SWH, the ANFIS model33 the ANFIS model incorporates three consecutive time steps of MSLP, SWS, and SWH as input features. This design explicitly accounts for the temporal dependencies inherent in ocean dynamics, enabling the model to consider the influence of past conditions when predicting future wave heights. ANFIS is a sophisticated hybrid computational framework34 that combines the adaptive learning capabilities of ANNs with the interpretability of fuzzy logic systems. As illustrated in Fig. 3, this integration allows the model to process the sequential time-lagged meteorological inputs. The schematic highlights the hybrid architecture, showing how the neural network component dynamically refines fuzzy rules using learning algorithms such as gradient descent or backpropagation.

To capture the nonlinear relationships between meteorological variables and SWH, the ANFIS model utilized three consecutive time steps of mean sea level pressure (MSLP), surface wind speed (SWS), and historical SWH as input features. This design effectively accounts for temporal dependencies inherent in oceanic dynamics, enabling accurate prediction of SWH at the next time step. The Adaptive Neuro-Fuzzy Inference System (ANFIS), combining the learning capabilities of neural networks with the interpretability of fuzzy logic, was implemented through a five-layer architecture (Fig. 3). To automate the generation of fuzzy inference rules, subtractive clustering was applied with a cluster radius of 0.5, selected through extensive experimentation with radii ranging from 0.3 to 0.8. This radius was determined based on minimizing the mean squared error (MSE) on a validation set while balancing model complexity and predictive accuracy. A radius of 0.5 was chosen for its optimal number of clusters, reducing overfitting while maintaining interpretability. Other hyperparameters, including the number and shape of membership functions (Gaussian with tuned mean and standard deviation), were optimized via grid search and 10-fold cross-validation to ensure model generalization. Training employed a hybrid learning algorithm: in the forward pass, linear consequent parameters were optimized using least-squares estimation; in the backward pass, nonlinear membership parameters were adjusted via gradient descent to minimize MSE. Early stopping based on validation error was implemented to prevent overfitting. This architecture was implemented using MATLAB’s Fuzzy Logic Toolbox, facilitating precise parameter tuning and performance evaluation.

Fig. 3
figure 3

Diagram of the ANFIS model architecture.

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Figure 3 provides a simplified diagram of the ANFIS architecture. The ANFIS network comprises five layers:

  1. 1.

    Fuzzification Layer: Each input (three consecutive time steps of MSLP, SWS, and SWH) is processed by Gaussian membership functions (MF) whose parameters (mean and standard deviation) are initialized via subtractive clustering (cluster radius = 0.5).

  2. 2.

    Rule Layer: Nodes represent fuzzy “if–then” rules; each rule’s firing strength is computed as the product of corresponding input membership degrees.

  3. 3.

    Normalization Layer: Firing strengths are normalized by the sum of all rule strengths.

  4. 4.

    Consequent Layer: Each normalized strength multiplies a linear function of inputs, with coefficients determined during training.

  5. 5.

    Output Layer: Aggregates all rule outputs to produce the final SWH prediction.

  6. 6.

    Training employs a hybrid learning algorithm: in the forward pass, linear consequent parameters are optimized by least-squares estimation; in the backward pass, nonlinear membership parameters are tuned via gradient descent to minimize Mean Squared Error (MSE).

Figure 4 presents a detailed schematic of the ANFIS multi-layer architecture. The black nodes on the left represent the input variables. The signal propagates from left to right through several distinct layers: (1) white nodes that correspond to MFs, where each input is fuzzified; (2) the rule layer composed of blue nodes, where fuzzy “if-then” rules to combine the fuzzified inputs; (3) output MFs (white nodes), which process and normalize the strengths of the firing rules; and (4) the defuzzification layer (white node[s]), which generates the overall system output.

The legend in the lower-right corner utilizes color coding to depict logical operations such as “and,” “or,” and “not,” providing a clear visual representation of how fuzzy logic is embedded within the neural network framework. This architectural design exemplifies the seamless integration of fuzzy logic with neural network learning, enabling ANFIS to perform efficient nonlinear modeling and inference.

Fig. 4
figure 4

Diagram of the multi-layer ANFIS architecture.

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The model’s input variables were selected to capture key temporal dependencies by including three consecutive time steps of MSLP, SWS, and SWH. Fuzzy inference rules were generated via subtractive clustering with a cluster radius of 0.5, striking an effective balance between model complexity and interpretability. The ANFIS framework was implemented using MATLAB’s Fuzzy Logic Toolbox35 which provides an accessible environment for parameter tuning and optimization.

The system is designed to predict SWH at the next time step by leveraging historical time-lagged data, thereby incorporating temporal dynamics into the forecasting process. To ensure effective training and generalization of the ANFIS model, several training strategies were employed. The loss function used during the model training phase was the MSE, as it is well-suited for continuous output regression tasks like wave height prediction. To prevent overfitting, early stopping was implemented by monitoring the validation error across epochs and terminating training when the error plateaued or began increasing. Additionally, 10-fold cross-validation was used during model development to ensure robustness. Hyperparameters—such as the number of MFs, the shape of the MFs (Gaussian), and the cluster radius for subtractive clustering (set to 0.5)—were tuned through iterative experimentation and grid search using MATLAB’s Fuzzy Logic Toolbox.

Model evaluation metrics

Key evaluation metrics—including root mean square error (RMSE), mean absolute error (MAE), and the coefficient of determination (R²), which corresponds to the square of the Pearson correlation coefficient—were employed to comprehensively assess the performance of the ANFIS model. These metrics quantify different aspects of the model’s accuracy in predicting SWH. The mathematical definitions of these metrics are provided in the following equations:

$$RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}{({y}_{i}-{\widehat{y}}_{i})}^{2}}$$
(1)
$$MAE=\frac{1}{n}\sum_{i=1}^{n}\left|{y}_{i}-{\widehat{y}}_{i}\right|$$
(2)
$${R}^{2}=\sum_{i=1}^{n}{(y}_{i}-\stackrel{-}{y}\left)\right({\widehat{y}}_{i}-\stackrel{-}{\widehat{y}})/\sum_{i=1}^{n}{{(y}_{i}-\stackrel{-}{y})}^{2}\times \sum_{i=1}^{n}{({\widehat{y}}_{i}-\stackrel{-}{\widehat{y}})}^{2}$$
(3)

where \({y}_{i}\)​ is the observed SWH, \({\widehat{y}}_{i}\)​ is the predicted SWH, and \({\widehat{y}}_{i}\)​, \(\stackrel{-}{\widehat{y}}\)​ are their respective means. These metrics were calculated for both the interpolation test set (30% of the data) and the extrapolation set consisting of 250 previously unseen records, enabling evaluation of the model’s generalization performance both within and beyond the training domain.

Result

Figure 5 illustrates the evolution of MFs in the final ANFIS model. The figure consists of multiple small subplots arranged in a grid, each depicting a Gaussian-shaped membership function with yellow-shaded areas indicating activated regions. The x-axis represents input variable values, while the y-axis shows the corresponding of membership degrees. Different training phases are visualized, demonstrating how the MFs adapt and refine through successive iterations. The rightmost columns correspond to the final training stage, where convergence of the membership functions has been achieved, reflecting the model’s stable and optimized fuzzy partitions.

Fig. 5
figure 5

Visualization of membership function evolution in the final ANFIS model.

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The converged Gaussian MFs in Fig. 5 reveal how ANFIS partitions each input’s domain based on data-driven clusters. For example, the first MSLP MF exhibits a narrow peak around lower pressures (~ 1008 hPa), capturing storm-like conditions, while a broader MF centered near typical sea‐level pressure (~ 1015 hPa) models normal variability. Similarly, the SWS MFs converge to cover low (< 10 m/s), moderate (10–20 m/s), and high (> 20 m/s) regimes, aligning with calm, windy, and gale conditions respectively. The lagged SWH MFs reflect that past wave heights below 1 m and above 2 m have distinct rule activations, indicating the model’s sensitivity to both calm sea states and extreme waves. These final shapes thus encode physically meaningful regimes, allowing ANFIS to adapt its rules to different ocean‐atmospheric scenarios.

To benchmark the performance of the ANFIS model, a diverse set of regression models was selected based on their frequent use in oceanographic and time-series prediction tasks. These include multilayer perceptrons (MLPs), SVMs, Gaussian process regression (GPR), decision trees, linear regression variants, and kernel-based regressors. This selection captures a broad spectrum of model complexity, generalization ability, and interpretability. MLPs and SVMs are widely applied in ocean wave prediction due to their nonlinear modeling capability, while GPR provides probabilistic insights and uncertainty quantification. Tree-based models offer fast and interpretable alternatives, and various linear regression techniques were included for their baseline value and computational efficiency. The models were trained using MATLAB’s Regression Learner App, where hyperparameter tuning was carried out using automatic Bayesian optimization or grid search depending on the model type. All models used standardized input data and were configured with default regularization and kernel settings unless otherwise specified (see Tabel 2).

A comparative analysis of the ANFIS model performance against the aforementioned models during the training, testing, and prediction phases is presented in Table 1. The hyperparameter configurations for all evaluated models are detailed in Table 2, providing insight into the factors contributing to the performance differences observed in Table 1. The results highlight that ANFIS maintains competitive accuracy compared to conventional ML approaches, exhibiting superior performance in modeling and interpolation (as indicated by RMSE and MAE values), although it demonstrates slightly higher errors during extrapolation. The high R² values further confirm the model reliability in capturing complex relationships within the dataset.

Table 1 Performance comparison: ANFIS model versus benchmark regression models.
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Table 2 Hyperparameter configurations for evaluated models.
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The comparative analysis reveals that the ANFIS model achieves strong predictive accuracy, while several neural network and Gaussian process regression models exhibit competitive generalization performance under specific conditions. For instance, the stepwise linear regression model closely matches the ANFIS performance during interpolation, attaining an R² value of 0.9091 on the testing dataset. In contrast, conventional regression models, including standard linear regression and tree-based methods, display substantially higher error rates, particularly during the extrapolation phase, indicating their limited ability to generalize beyond the training domain.

Figures 6, 7 and 8 provide a comprehensive evaluation of the ANFIS model’s performance across the training, testing, and prediction phases, respectively. Figure 6 illustrates the time-series and scatter plot comparisons of observed versus predicted SWH during training, demonstrating low error (RMSE = 0.3897 m, R²=0.9185) and close alignment with the 1:1 reference line, confirming the reliability of the training process. Figure 7 depicts the model’s performance on the testing dataset (RMSE = 0.3808 m, R²=0.9089), validating its generalization capability within observed data distributions. Figure 8 showcases forecasting performance on entirely unseen data (250 samples), achieving an RMSE of 0.5142 m and R² of 0.8948, affirming the model’s robust predictive ability for operational applications. The superior performance of the ANFIS model compared to benchmark models (Table 1) stems from its use of subtractive clustering, which automatically extracts fuzzy rules from the data, effectively modeling nonlinear relationships among input variables (MSLP, SWS, and SWH). Additionally, incorporating three consecutive time-lagged inputs captures temporal wave dynamics, enhancing prediction accuracy, particularly in the variable conditions of the Southern Ocean. The slight increase in error during the prediction phase (MAPE = 8.63%) is attributable to data distribution shifts in the prediction set, potentially involving unseen atmospheric or oceanic conditions, such as extreme winds. These results, combined with a rapid prediction time of 65 milliseconds per sample, highlight the ANFIS model’s strong potential for operational SWH forecasting in challenging polar environments.

Fig. 6
figure 6

ANFIS model performance during training phase: (a) Time-series comparison of observed and predicted SWH; (b) Scatter plot of observed versus predicted SWH, with a 1:1 reference line.

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Fig. 7
figure 7

ANFIS model performance during testing phase: (a) Time-series comparison of observed and predicted SWH; (b) Scatter plot of observed versus predicted SWH, with a 1:1 reference line.

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Fig. 8
figure 8

ANFIS model performance during prediction: (a) Time-series comparison of observed and predicted SWH; (b) Scatter plot of observed versus predicted SWH, with a 1:1 reference line.

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To assess the operational feasibility of the proposed ANFIS model, we measured both the total training time and average prediction speed on a workstation equipped with an Intel Core i5-12450 H CPU (2.00 GHz) and 16 GB of RAM, running MATLAB R2024a. The model required 3.59 s to complete training under the described subtractive clustering and convergence criteria. Following training, the model achieved an average prediction time of 65 milliseconds per sample, demonstrating its capability for near real-time forecasting applications.

Discussion

The findings of this study affirm the capacity of the ANFIS model to effectively represent nonlinear relationships between meteorological variables and SWH. The observed dominance of SWS as the most influential factor aligns consistently with established wave generation theory. Furthermore, the integration of time-lagged features significantly enhances prediction accuracy, underscoring the critical importance of accounting for the temporal evolution of ocean wave behavior. A detailed examination of Table 1 provides further insights into the model’s performance. During the training and testing phases, the ANFIS model exhibits low RMSE values (0.3897 and 0.3880 m, respectively) and MAE values (0.2740 and 0.2822 m, respectively), accompanied by high R² coefficients (0.9185 for training and 0.9089 for testing). These metrics indicate that the model effectively learns the underlying data relationships and provides consistent predictions within the range of the training data. While a modest increase in RMSE (0.5142 m) and a slight decrease in R² (0.8948) are observed in the extrapolation phase, the model’s continued competitive performance suggests a reasonable degree of generalization when predicting entirely unseen data, albeit with slightly reduced accuracy. Comparative analysis reveals that while some neural network configurations and Gaussian process regression models achieve accuracy comparable to that of ANFIS, they often do so at the expense of increased complexity and computational requirements. For example, certain neural network models demonstrate similar training and testing performance to ANFIS; however, their deeper architectures can limit their applicability in real-time settings due to longer training durations and higher demands on computational resources. Similarly, although specific support vector machine variants (e.g., SVM 2 and SVM 6) closely approach the performance of ANFIS in the initial training and testing phases, they tend to exhibit more pronounced error increases during extrapolation, indicating limitations in their ability to generalize to unseen data.

The consistent performance of the ANFIS model across all three phases—training, testing, and extrapolation—underscores the benefits of its specific architecture in managing nonlinear dynamics. The synergistic combination of adaptive fuzzy inference, subtractive clustering, and time-lagged inputs allows the model to effectively address both the intricacies of model training and the challenges inherent in forecasting future trends. In contrast, simpler models like linear regression, while computationally efficient, demonstrably lack the capacity to capture these complex interactions and exhibit higher error rates, especially during extrapolation.

ANFIS outperformed most benchmark models; however, stepwise linear regression produced comparable results. This similarity in performance likely stems from the relatively strong linear relationships identified between SWH and key input variables such as SWS and MSLP, as demonstrated in the correlation analysis (Fig. 1). Although simpler in structure, stepwise linear regression leverages this linearity effectively and further benefits from its capability to exclude irrelevant predictors, thereby reducing the risk of overfitting. Additionally, given that the forecasting task relies on a limited set of strongly correlated meteorological predictors, a straightforward linear model can approximate the output with similar accuracy to more complex nonlinear models, particularly in interpolation contexts. As shown in the extrapolation results (Fig. 8), all models experienced a decline in accuracy when evaluated on the extrapolation dataset, which was anticipated. This reduction in performance is primarily attributable to a distribution shift between the training data and the extrapolation set. Specifically, the extrapolation records correspond to future time periods that may encompass atmospheric and oceanic conditions not fully represented during training—for example, extreme winds or anomalous pressure gradients. Moreover, the extrapolation dataset is relatively compact (consisting of 250 records), potentially limiting variability and exacerbating class imbalance or underrepresentation of rare but significant patterns. These factors collectively contribute to diminished generalization and increased errors across all models. Nevertheless, the ANFIS model demonstrated comparatively better resilience due to its capacity for nonlinear adaptation.

The robust performance of the ANFIS model across modeling, interpolation, and extrapolation tasks—as demonstrated in Table 1—validates the effectiveness of this approach for capturing nonlinear dynamics in wave height prediction. Future research should investigate hybrid frameworks that integrate fuzzy logic with advanced deep learning techniques to further improve predictive accuracy in maritime forecasting applications.

To facilitate adoption within existing maritime forecasting workflows, the proposed ANFIS model can be encapsulated as a standalone module that interfaces via standard application programming interfaces (APIs) or common data formats such as NetCDF and GRIB2. In operational settings, the trained ANFIS engine would receive real-time input streams of MSLP and SWS from meteorological data providers such as ECMWF or NOAA, and generate one-step-ahead wave height forecasts in either batch or streaming modes. Integration into end-to-end forecasting systems can be achieved by embedding the MATLAB-exported model as a compiled function using tools like MATLAB Compiler SDK, or by re-implementing the fuzzy inference rules in languages such as Python or Java to run on high-performance servers. This architecture enables seamless interoperability with pre-existing components—such as data assimilation, post-processing, and visualization modules—allowing end users, including ship routing services and polar research platforms, to incorporate ANFIS forecasts alongside conventional numerical outputs without disrupting current operational pipelines.

Conclusion

This study demonstrates that an ANFIS optimized with subtractive clustering achieves highly accurate SWH forecasting in the SO, with a RMSE ranging from 0.3897 to 0.5142 m and R² up to 0.8948, indicating robust accuracy and generalization in challenging polar environments. Key innovations include (1) the integration of sequential time-lagged inputs to capture wave dynamics, (2) the use of subtractive clustering to automatically generate interpretable fuzzy rules, and (3) computational efficiency (65 ms prediction time) suitable for operational applications. These features position the ANFIS model as a powerful tool for wave forecasting in data-sparse regions, enhancing maritime safety and advancing understanding of ocean-atmosphere interactions. However, as noted by Guisado-Pintado, the slight reduction in accuracy during extrapolation highlights the need for improved generalization11. For future research, we recommend (1) incorporating additional data sources, such as sea surface temperature and ocean currents, to cover a broader range of environmental conditions, (2) exploring hybrid architectures with deep learning models like LSTM to enhance modeling of complex dynamics, and (3) developing transfer learning techniques to improve accuracy for rare and unseen conditions. These advancements could further elevate the accuracy and reliability of SWH forecasting in polar regions, expanding its operational applicability.