Systematic Comparison of Physics-Informed and Data-Driven Neural Networks for Solving the 1D Schrödinger Equation

The prediction of electronic properties from quantum mechanical principles is a corner- stone of computational chemistry and materials science. This paper presents a systematic comparison between Physics-Informed Neural Networks (PINNs) and standard Data-Driven Neural Networks (DDNNs) for solving the one-dimensional time-independent Schrödinger equation, a fundamental model for electronic structure prediction. We evaluate both paradigms across multiple potentials—including the infinite square well, quantum harmonic oscillator, and anharmonic (quartic) oscillator—assessing accuracy, data efficiency, generalization capability, and computational cost. Our results demonstrate that PINNs achieve superior accuracy in low-data regimes and extrapolate robustly beyond the training domain by embedding the governing differential equation into the loss function. Conversely, DDNNs exhibit faster training convergence and higher accuracy when abundant high-fidelity training data is available. The quantum harmonic oscillator case reveals that PINNs achieve relative eigenvalue errors below 10^−5 with minimal training data, while DDNNs require approximately 100× more data points to reach comparable precision. For the anharmonic oscillator, where no closed-form solution exists, PINNs maintain physical consistency (normalization, boundary conditions) even with noisy data, whereas DDNNs suffer from systematic drift. This study provides actionable guidelines for selecting the appropriate neural network paradigm based on data availability, problem complexity, and physical constraint requirements in quantum mechanical simulations.