The effective estimation of mixed-layer depth (MLD) plays a significant role in the study of ocean dynamics and global climate change. However, the methods of estimating MLD still have limitations due to the sparse resolution of the observed data. In this study, a hybrid estimation method that combines the K-means clustering algorithm and an artificial neural network (ANN) model was developed using sea-surface parameter data in the Indian Ocean as a case study. The oceanic datasets from January 2012 to December 2019 were obtained via satellite observations, Argo in situ data, and reanalysis data. These datasets were unified to the same spatial and temporal resolution (1° × 1°, monthly). Based on the processed datasets, the K-means classifier was applied to divide the Indian Ocean into four regions with different characteristics. For ANN training and testing in each region, the gridded data of 84 months were used for training, and 12-month data were used for testing. The ANN results show that the optimized NN architecture comprises five input variables, one output variable, and four hidden layers, each of which has 40 neurons. Compared with the multiple linear regression model (MLR) with a root-mean-square error (RMSE) of 5.2248 m and the HYbrid-Coordinate Ocean Model (HYCOM) with an RMSE of 4.8422 m, the RMSE of the model proposed in this study was reduced by 27% and 22%, respectively. Three typical regions with high variability in their MLDs were selected to further evaluate the performance of the ANN model. Our results showed that the model could reveal the seasonal variation trend in each of the selected regions, but the estimation accuracy showed room for improvement. Furthermore, a correlation analysis between the MLD and input variables showed that the surface temperature and salinity were the main influencing factors of the model. The results of this study suggest that the pre-clustering ANN method proposed could be used to estimate and analyze the MLD in the Indian Ocean. Moreover, this method can be further expanded to estimate other internal parameters for typical ocean regions and to provide effective technical support for ocean researchers when studying the variability of these parameters.

The recent success of emission reduction policies in China has significantly lowered sulfur dioxide (SO₂) levels. However, accurately forecasting these concentrations remains challenging due to their inherent non-stationary tendency. This study introduces an innovative hybrid deep learning model, RF-VMD-Seq2Seq, combining the Random Forest (RF) algorithm, Variational Mode Decomposition (VMD), and the Sequence-to-Sequence (Seq2Seq) framework to improve SO₂ concentration forecasting in five coastal cities of northern China. Our results show that the predicted SO₂ concentrations closely align with observed values, effectively capturing fluctuations, outliers, and extreme events—such as sharp declines the Novel Coronavirus Pneumonia (COVID-19) pandemic in 2020—along with the upper 5% of SO₂ levels. The model achieved high coefficients of determination (>0.91) and Pearson’s correlation (>0.96), with low prediction errors (RMSE < 1.35 μg/m³, MAE < 0.94 μg/m³, MAPE < 15%). The low-frequency band decomposing from VMD showed a notable long-term decrease in SO₂ concentrations from 2013 to 2020, with a sharp decline since 2018 during heating seasons, probably due to the ‘Coal-to-Natural Gas’ policy in northern China. The input sequence length of seven steps was recommended for the prediction model, based on high-frequency periodicities extracted through VMD, which significantly improved our model performance. This highlights the critical role of weekly-cycle variations in SO₂ levels, driven by anthropogenic activities, in enhancing the accuracy of one-day-ahead SO₂ predictions across northern China’s coastal regions. The results of the RF model further reveal that CO and NO₂, sharing common anthropogenic sources with SO₂, contribute over 50% to predicting SO₂ concentrations, while meteorological factors—relative humidity (RH) and air temperature—contribute less than 20%. Additionally, the integration of VMD outperformed both the standard Seq2Seq and Ensemble Empirical Mode Decomposition (EEMD)-enhanced Seq2Seq models, showcasing the advantages of VMD in predicting SO₂ decline. This research highlights the potential of the RF-VMD-Seq2Seq model for non-stationary SO₂ prediction and its relevance for environmental protection and public health management.

Physics–Informed Neural Networks (PINNs) have demonstrated efficacy in solving both forward and inverse problems for nonlinear partial differential equations (PDEs). However, they frequently struggle to accurately capture multiscale physical features, particularly in regions exhibiting sharp local variations such as shock waves and discontinuities, and often suffer from optimization difficulties in complex loss landscapes. To address these issues, we propose LocRes–PINN, a physics–informed neural network framework that integrates local awareness mechanisms with residual learning. This framework integrates a radial basis function (RBF) encoder to enhance the perception of local variations and embeds it within a residual backbone to facilitate stable gradient propagation. Furthermore, we incorporate a residual–based adaptive refinement strategy and an adaptive weighted loss scheme to dynamically focus training on high–error regions and balance multi–objective constraints. Numerical experiments on the Extended Korteweg–de Vries, Navier–Stokes, and Burgers equations demonstrate that LocRes–PINN reduces relative prediction errors by approximately 12% to 67% compared to standard benchmarks. The results also verify the model’s robustness in parameter identification and noise resilience.