Comparative Study between Numerical Solver, Analytical Perturbation Theory and Physics-Informed Neural Networks in solving a quartic Anharmonic Oscillator

Abstract Perturbation theory is a widely used analytical technique for estimating shifts in the energy levels of atoms when a perturbation is introduced into a system. However, as demonstrated in this study, the resulting Rayleigh–Schrödinger perturbation series is divergent for every non-zero value of the perturbation parameter λ. In this paper, we develop and compare three approaches for solving the same perturbed Hamiltonian problem: Hamiltonian equation: H = -(1/2)(d²/dx²) + (1/2)x² + λx⁴ The three approaches are: 1. Exact Numerical Solver: Uses finite-difference discretization of the time-independent Schrödinger equation, followed by Hamiltonian diagonalization. The results are cross-validated against an independent Fock-basis spectral solver. 2. Analytical Perturbation Series: Derived using ladder operators to calculate energy corrections. 3. Physics-Informed Neural Network (PINN): Uses a neural network to approximate the wavefunctions and corresponding energy eigenvalues. The results are compared as the perturbation parameter λ increases from 0 to 1. This allows us to identify precisely where perturbation theory becomes unreliable and evaluate the accuracy of PINN-based solutions. Third-order perturbation theory is exact at λ = 0, but its error reaches 18.6 energy units at λ = 1, which is more than twenty times larger than the quantity being estimated. Across 80 cold-start training runs (five random seeds at eight values of λ), the dedicated PINN achieves mean energy errors of: - Ground state: 2.87 × 10⁻⁵ - First excited state: 3.62 × 10⁻⁵ Further extracting the energy from the trained wavefunction using the Rayleigh quotient reduces these errors to: - Ground state: 6.92 × 10⁻⁶ - First excited state: 2.54 × 10⁻⁶ At the points tabulated by Brevi et al., the relative energy error is 28.8 to 1.08 × 10³ times lower for states n ≤ 2, although sequential training fails for states n ≥ 3. Furthermore, we develop a parametric PINN, in which a single neural network takes both x and λ as inputs and learns ψ(x; λ) and E(λ) over the entire interval λ ∈ \[0, 1\] in a single training run. This implements the lambda-as-input extension proposed as future work by Brevi et al. The parametric PINN achieves an energy-network accuracy of 2.46 × 10⁻³ for λ greater than 0. However, extracting energies from its learned wavefunctions using the Rayleigh quotient improves the accuracy to 2.59 × 10⁻⁵. These results demonstrate that the PINN successfully generalizes across different perturbation strengths, although its direct energy prediction is less accurate than its wavefunction-based energy estimation.