Spectral, spectral element, and hp finite element methods are central high-order tools for high-fidelity scientific computing. Their high accuracy, tensor-product structure, locality, and high computation-to-memory ratio make them well suited for modern heterogeneous and exascale architectures. Beyond accuracy, however, reliable simulations of complex physical systems require the preservation of intrinsic structures such as conservation laws, divergence constraints, positivity, entropy dissipation, symplecticity, and energy stability. The rise of AI for Science offers new opportunities to accelerate and augment high-order methods through neural operators, physics-informed learning, learned solvers, data-driven closures, and AI-assisted adaptivity. Yet robustness, physical consistency, and long-time stability remain major challenges for purely data-driven models. This minisymposium focuses on structure-preserving and high-fidelity spectral/high-order methods in the era of AI4S, emphasizing rigorous numerical foundations, efficient HPC implementations, and principled integration with machine learning. Topics include conservative, divergence-free, positivity-preserving, entropy-stable and energy-stable schemes; asymptotic-preserving and symplectic methods; fast solvers, preconditioners, and GPU-oriented implementations; spectral or polynomial-basis neural operators; variational and physics-informed learning; learned hp-adaptivity; AI-assisted preconditioning; and structure-preserving neural closures, with applications in fluids, electromagnetics, kinetic equations, waves, materials, geophysical flows, and multiphysics simulations.