A Comprehensive Survey of Non-Euclidean Domain Applications in Diverse Mechanical Designs

The increasing importance of non-Euclidean geometry in machine learning and mechanical engineering is examined in this paper. Modern engineering systems and data structures frequently contain complex, curved, or high-dimensional areas that are difficult for traditional Euclidean models to handle. For such problems, non-Euclidean frameworks provide more accurate and adaptable representations, particularly Riemannian and hyperbolic geometries. They make it possible to precisely simulate materials and structures with inherent curvature in mechanical engineering, which helps with the creation of smart and bioinspired materials. Non-Euclidean methods in machine learning improve neural network performance, especially in tasks involving classification, reinforcement learning, and graph-based learning.The theoretical underpinnings, important computational models, and practical applications spanning from autonomous systems to structural mechanics are highlighted in this overview. Non-Euclidean approaches offer a potent paradigm for solving spatial complexity challenges by fusing geometry and computation, opening up new avenues for interdisciplinary innovation.