Online Identification Method of Orbital Angular Momentum in Ocean Turbulent Environment Based on OS-ELM

Orbital Angular Momentum （OAM） beams， featuring spiral phase wavefronts， act as critical new information carriers in optical communication， manipulation， imaging and quantum information. However， ocean turbulence， driven by temperature and salinity fluctuations， induces intensity scintillation， phase distortion and mode crosstalk in OAM beams， hindering their marine application. Traditional identification methods relying on optical components are complex and environment-sensitive， while current machine learning-based static models fail to satisfy marine real-time response requirements due to high retraining costs.To address these challenges， this study proposes an online OAM identification method based on the Online Sequential Extreme Learning Machine （OS-ELM）. This efficient online sequential learning algorithm has a three-layer structure. The input layer receives feature vectors of OAM beams， with the number of nodes matching the feature dimension. The hidden layer has a fixed number of nodes and supports two activation modes： additive mode and RBF-type mode. Additive mode uses random weights and biases， while RBF-type mode uses random centers and influence factors. The parameters of the hidden layer are fixed after random initialization， eliminating the need for time-consuming gradient iterations. The output layer generates One-Hot encoded labels for OAM modes， and the output weights are solved through Moore-Penrose generalized inverse to ensure fast model training. The implementation of the method in this study consists of two key components： data generation and model construction. In the data generation phase， a numerical simulation model is built using MATLAB to generate an OAM spot dataset affected by oceanic turbulence. Key parameters include the specific inner and outer scales of oceanic turbulence， an appropriate waist radius of Laguerre-Gaussian beams， a wavelength commonly used in optical communications， and fixed spacing of random phase screens. The dataset covers LG modes with topological charges ranging from 1 to 10 and generates intensity images. For data preprocessing， 200 turbulence-affected images are generated for each topological charge， forming a training set of 2 000 images and a test set of 600 images. After adjusting the image size via bicubic interpolation， performing grayscale conversion， double-precision conversion， and normalization to the range of ［-1， 1］， the 2D image matrices are flattened into 1D feature vectors， which are combined with labels to form a feature matrix. In the model construction phase， the dataset is divided into an initial training block with 1 000 samples and 20 online training blocks with 50 samples each. The initial training block is used to calculate the hidden layer output matrix， solve the initial weights by combining the label matrix and Moore-Penrose generalized inverse， and define an intermediate matrix. During online learning， the hidden layer output matrix of the new data block is first computed； the intermediate matrix is then updated using the Woodbury identity without retraining historical data. Finally， the output weights are incrementally updated using the new labels and prediction residuals， ensuring the model dynamically adapts to OAM beam feature drift while maintaining real-time performance.Based on the same dataset， comprehensive experimental comparisons are conducted between the Online Sequential Extreme Learning Machine OS-ELM and other models， including Support Vector Machine （SVM）， Incremental Naive Bayes classifier （IncNB）， and Naive Bayes classifier （NB）， with performance evaluated using multiple metrics. In weak turbulence scenarios， the Accuracy of OS-ELM reaches 92.33% higher than SVM's 85.33%， IncNB's 87.00%， and NB's 88.67%， with simultaneous leadership in Precision， Recall， and F1-Score. In moderate turbulence， its Accuracy is 71.17% far exceeding SVM's 49.50%， IncNB's 51.83%， and NB's 62.00%. Even in strong turbulence， it maintains an Accuracy of 54.33% higher than SVM's 37.33%， IncNB's 38.83%， and NB's 50.67%. The ROC （Rate of Change） curve of OS-ELM clusters near the top-left corner of the coordinate plane， with AUC （Area Under Curve） values of 0.992， 0.946， and 0.890 under weak， moderate， and strong turbulence respectively， significantly outperforming the comparison models.The advantages of OS-ELM stem from two core mechanisms： first， the random initialization of hidden layer parameters and analytical solution of output weights avoid the risk of falling into local optima and eliminate the need for complex kernel function calculations required by SVM， enabling efficient approximation of the global optimal solution； second， the online sequential learning mechanism supports real-time incremental updates， accurately adapting to OAM beam feature drift caused by oceanic turbulence and overcoming the inherent limitation of traditional static models with fixed parameters. Looking ahead， the dynamic adaptability， high robustness， and real-time performance of the online sequential extreme learning machine make it highly compatible with complex interference and high-frequency dynamic changes in marine environments. With further engineering development， this method holds significant potential to drive technological upgrades in intelligent manufacturing and industrial communication.