Explainable artificial intelligence and mathematics: What lies behind? Let us focus on this new research field

The growing complexity inherent in modern artificial intelligence (AI) models has necessitated an increased focus on the demand for explainability, commonly referred to as explainable artificial intelligence (XAI). The primary objective of XAI is to render the decision-making processes of AI systems not only transparent, but also understandable to human users, thereby fostering greater trust and comprehension among stakeholders. As AI systems become more sophisticated and are deployed in critical areas such as healthcare, finance, and autonomous vehicles, the demand for clarity surrounding their operations intensifies. This paper delves deeply into the vital relationship between XAI and mathematics, asserting that mathematical principles are foundational to enhancing the interpretability, transparency, and overall trustworthiness of AI models. We will investigate the key mathematical constructs that underlie various XAI techniques, providing insights into how they function and contribute to explainability. To illustrate the practical significance of these principles, we will examine specific case studies where mathematical frameworks have successfully improved the elucidation of AI model predictions. Furthermore, this paper will outline potential future avenues for research that aim to further integrate mathematical methodologies within XAI frameworks. By doing so, we hope to contribute to the development of more robust and interpretable AI systems that can be trusted and effectively utilized by humans in a multitude of applications.