Mathematical Foundations of Intelligent Systems: Artificial Intelligence and Machine Learning under Uncertainty Using Fuzzy Numbers

Artificial Intelligence (AI) and Machine Learning (ML) systems rely on data-driven mathematical models for representation, prediction, and inference; however, real-world data is often affected by epistemic uncertainty, vagueness, and non-stochastic imprecision that cannot be adequately captured by classical probabilistic approaches. This paper proposes a fuzzy-intelligent regression framework that integrates classical linear regression with fuzzy set theory to model uncertainty in a more expressive manner. A synthetic dataset generated from a linear stochastic process with Gaussian noise is used, and a baseline least-squares regression model is developed. To incorporate uncertainty, input features are represented as triangular fuzzy numbers, enabling interval-based propagation through the regression function and producing bounded predictions. Additionally, a fuzzy membership function is defined to quantify the degree of compatibility of inputs within their uncertainty ranges, providing an interpretable measure of vagueness beyond statistical variance. The model is evaluated using mean squared error (MSE) and a robustness index based on residual dispersion. Results show that while classical regression captures the central trend effectively, the fuzzy-enhanced model improves interpretability by explicitly representing uncertainty bounds and membership grades, offering a computationally efficient approach for uncertainty-aware learning in AI/ML systems.