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.

Model-based parameter estimation, identification, and optimisation play a dominant role in many aspects of physical and operational processes in applied sciences, engineering, and other related disciplines. The intricate task involves engaging and fitting the most appropriate parametric model with nonlinear or linear features to experimental field datasets priori to selecting the best optimisation algorithm with the best configuration. Thus, the task is usually geared towards solving a clear optimsation problem. In this paper, a systematic-stepwise approach has been employed to review and benchmark six numerical-based optimization algorithms in MATLAB computational Environment. The algorithms include the Gradient Descent (GRA), Levenberg-Marguardt (LEM), Quasi-Newton (QAN), Gauss-Newton (GUN), Nelda-Meald (NEM), and Trust-Region-Dogleg (TRD). This has been accomplished by engaging them to solve an intricate radio frequency propagation modelling and parametric estimation in connection with practical spatial signal data. The spatial signal data were obtained via real-time field drive test conducted around six eNodeBs transmitters, with case studies taken from different terrains where 4G LTE transmitters are operational. Accordingly, three criteria in connection with rate of convergence Results show that the approximate hessian-based QAN algorithm, followed by the LEM algorithm yielded the best results in optimizing and estimating the RF propagation models parameters. The resultant approach and output of this paper will be of countless assets in assisting the end-users to select the most preferable optimization algorithm to handle their respective intricate problems.