Advancing Boole's rule inequalities through fractal analysis and neural network modeling

The goal of this study is to improve some known results related to Boole's type inequalities that use five points (Boole's rule). We first prove an important auxiliary identity connected to these inequalities. Using this auxiliary identity, we develop new Boole's type inequalities by applying a differentiable convex function within the setting of local fractional calculus. In this work, we study different types of functions, including convex, bounded, and Lipschitz functions over fractal sets. Additionally, a feedforward Artificial Neural Network (ANN) was used to approximate the left-hand side and right-hand side of fractal Boole-type inequalities. The model takes two input values and passes them through hidden layers to produce two outputs as predictions. This type of ANN is widely used because it can learn complex relationships from data without needing any fixed formulas. In this work, we apply an ANN model for the first time to predict the bounds of inequalities in fractal dimensions, which is an important outcome of our study. The ReLU activation function was applied to help the model learn nonlinear patterns, while training was carried out using the Mean Squared Error (MSE) loss and the Adam optimizer for stable and efficient learning. The network was trained for 500 epochs, and its performance was evaluated using loss curves. Finally, 3-dimensional surface plots were created to compare the predicted and actual inequality values. We also present examples and applications to show the usefulness of our main results.