## 1 Introduction
In particular, channel equalization techniques to combat interference introduced by linear or non-linear distortions are key to enable the correct re-construction of the transmitted symbols at the receiver. Several different signal equalization algorithms have been applied in UWOC systems to perform adaptive channel estimation in UWOC transmission channels, including least mean squares [19], linear minimum mean squared error [20], recursive least squares (RLS) [21], etc. In general, RLS has the best performance among them for UWOC channel estimation as it can effectively improve the signal-to-noise ratio by recursively updating the coefficients [25; 26]. Beside these conventional equalization algorithms, with the fast advancement of machine learning technologies in recent years, neural networks have been employed in optical communication networks [22; 23; 24; 27; 28; 29; 30] to assist the signal processing and enhance the system's ability to resist both linear and non-linear distortion. For example, in [29], an equalizer for an underwater visible light communication (UVLC) system was built by training a convolution-enhanced long short-term memory (CE-LSTM) neural network to approximate the correct mapping from delayed channel output to originally transmitted symbols. It had comparable efficiency and performances compared with a conventional Volterra-series-based equalizer and a general LSTM-based equalizer. Ref. [30] reported a multilayer perceptron (MLP) artificial neural network (ANN) classifier which was adopted as the equalizer to double the data rates in the visible light communications link. A recurrent neural network (RNN) was proposed in [23]. RNNs have been demonstrated to be superior to MLPs in learning non-linear mappings of arbitrary complexity in processing the time-series data. However, training RNNs is a non-linear nonconvex optimization problem. Stochastic gradient descent algorithms are computational demanding and prone to fall into local minima [31]. The exploding gradient problem can also cause unstable training and prediction performance [32]. Echo state networks (ESNs) are a type of reservoir computing (RC) which was proposed as an alternative, computationally efficient method to address these problems in RNNs [33; 34; 35; 36; 37]. ESNs can predict and classify multivariate time series by exploiting a sparsely connected hidden layer (the \"reservoir\") to create a higher-dimension representation (embedding) of their input. Their computational efficiency stems from the fact that the hidden layer is fixed, and only the weights of the connections between the hidden layer and the output need to be trained. Since the error function is quadratic with respect to the weights, the training does not suffer from vanishing/exploiting gradient problems. To further enhance the training and prediction performance of ESNs, a layered architecture, known as the deep echo state network (DeepESN), was proposed in 2017 [38]. It is uniquely suited to handle temporal data due to its structured state space organization with multi-timescale dynamics. This structure innately aligns with the compositional nature of recurrent neural modules involved in temporal data processing. DeepESNs exhibit an optimized effectiveness in layered recurrent neural networks [39; 40]. The DeepESN method has been improved and applied in multiple fields to solve different problems since its proposal [39; 40; 41; 42; 43]. For medical diagnosis, DeepESN can be used as a new method for the diagnosis of Parkinson's Disease (PD) [44]. In [45], based on time series analysis, a serially connected multi-reservoir echo state network (MR-ESN) is developed to conduct the prediction of solar irradiance, which has the advantages of an ESN, such as high efficiency. In short-term traffic prediction, the DeepESN is used to predict the future values of the traffic parameters of interest accurately [46]. Despite the proven ability of DeepESNs' ability to address time-series data across various domains, their application in the realm of UWOC remains unexplored.