Multi-Dimensional Quantum Anharmonic Oscillators via Physics-Informed Transformer Networks: Extension to Non-Perturbative Regimes and Higher Dimensions

This study extends the one-dimensional anharmonic oscillators by implementing physics-informed transformer networks (PINNs) for multi-dimensional quantum systems. We develop a novel computational framework that combines transformer architecture with physics-informed neural networks to solve the Schrodinger equation for 2D and 3D anharmonic oscillators, addressing both perturbative and non-perturbative regimes. The methodology incorporates attention mechanisms to capture long-range quantum correlations, orthogonal loss functions for eigenfunction discovery, and adaptive training protocols for progressive dimensionality scaling. Our approach successfully computes eigenvalues and eigenfunctions for quartic anharmonic oscillators in multiple dimensions with coupling parameters ranging from weak (λ = 0.01) to strong (λ = 1000) regimes. Results demonstrate superior accuracy compared to traditional neural networks, with mean absolute errors below 10-6 for ground state energies and the successful capture of symmetry breaking in anisotropic systems. The transformer-based architecture requires 60% fewer trainable parameters than conventional feedforward networks while maintaining comparable accuracy. Applications to molecular vibrational systems and solid-state physics demonstrate the practical utility of this approach for realistic quantum mechanical problems beyond the scope of perturbative methods.