High-order discretizations provide a powerful foundation for fast, efficient, and accurate scientific simulations on exascale and emerging computing architectures. At the same time, artificial intelligence and scientific machine learning are creating new opportunities to accelerate simulation, enhance solution accuracy, construct scalable surrogate models, and enable data-driven discovery and optimization. This minisymposium will highlight advances at the intersection of high-order numerical methods, performance-portable software,exascale computing, and AI-enabled scientific computing. Topics will include finite and spectral element algorithms; high-order computational kernels and lightweight portable libraries; scalable solvers and preconditioners; performance portability across heterogeneous architectures; and exascale applications. The minisymposium will also feature AI and machine-learning approaches that complement high-fidelity simulation, including physics-informed learning, neural operators, reduced-order and surrogate models, super-resolution methods, foundation models, and AI-assisted simulation, design, and control.