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There are many studies of approximations using deep neural networks. In this paper, the authors provide yet another proof that deep neural networks are universal approximators. In their earlier work, the authors showed that an arbitrary binary target function can be effectively rewritten in terms of a set of strings, or a set of subsets, and that a single hidden neuron can identify and only identify a single string or a single subset. Therefore, an arbitrary binary target function can be effectively rewritten in the form of a neural network with one hidden layer. In this study, the authors imposed locality on the deep neural network, and will show here that an arbitrary binary target function can be effectively rewritten in the form of a locally connected deep neural network that can have many hidden layers. Although it will increase the neural network size, as a neural network is localized, it will generally increase the speed of training for large networks 

This study developed a hybrid model for predicting dissolved oxygen (DO) using real-time sensor data for thirteen parameters. This novel hybrid model integrated one-dimensional convolutional neural networks (CNN) and long short-term memory (LSTM) to improve the accuracy of prediction for DO in water. The hybrid CNNLSTM model predicted DO concentration in water using soft sensor data. The primary input parameters to the model were temperature, pH, specific conductivity, salinity, density, chlorophyll, and blue-green algae. The model used 38,681 water quality data for training and testing the hybrid deep learning network. The training procedure for the model was successful. The training and test losses were both nearly zero and within a similar range. With a coefficient of determination (R2) of 0.94 and a mean squared error (MSE) of 0.12, the hybrid model indicated higher performance compared to the classical models. The normal distribution of residual errors confirmed the reliability of the DO predictions by the hybrid CNN-LSTM model. Feature importance analysis indicated pH as the most significant predictor and temperature as the second important predictor. The feature importance scores based on extreme gradient boosting (XGBoost) for the pH and temperature were 0.76 and 0.12, respectively. This study indicated that the hybrid model can outperform the classical machine learning models in the real-time prediction of DO concentration.