Application of Physics-Informed Neural Networks (PINNs) for the Numerical Solution of the Time-Independent Schrödinger Equation

Abstract

This work investigates the application of Physics-Informed Neural Networks (PINNs) for numerical solutions to the time-independent Schrödinger equation of the quantum harmonic oscillator in one, two, and three spatial dimensions. Fully connected neural architectures are constructed to approximate wavefunctions over finite symmetric domains, while the corresponding energy eigenvalues are treated as trainable parameters. The training strategy utilizes randomly sampled interior points to enforce the Schrödinger operator residual and boundary points to impose vanishing wavefunction constraints. For the 1D quantum harmonic oscillator, the learned ground-state wavefunction yields an energy of E = 1.2939 after 12,000 iterations. In the 2D configuration, convergence is achieved at E = 2.1352 within 14,000 iterations, whereas the 3D model attains E = 2.6377 after 12,000 iterations. These values agree with the expected trend of increasing ground-state energy with dimensionality, although deviations from exact analytical values indicate that PINNs may experience optimization challenges and sensitivity to sampling density and boundary enforcement. Despite these limitations, the trained models successfully capture the characteristic spatial symmetries and Gaussian-like envelope of harmonic oscillator eigenstates across all dimensions. These findings demonstrate that PINNs offer a flexible, mesh-free alternative for solving stationary quantum systems, particularly when analytical or conventional numerical approaches become impractical. The method shows strong potential for higher-dimensional quantum applications, even though further refinement such as improved sampling, loss balancing, and network depth remains necessary to suppress residual error and enhance eigenvalue accuracy.