Solution of Time-Dependent Schrödinger Equation using Physics-based Neural Network

Solving the time-dependent Schrodinger equation (TDSE) in the presence of noisy or incomplete data is a significant challenge in quantum mechanics, particularly when both the real and imaginary components of the wavefunction must be accurately reconstructed. In this work, we present a physics-informed neural network (PINN) framework to solve the TDSE for a one-dimensional quantum harmonic oscillator across multiple energy levels n = 0, 1, 2. The noisy input data used for training is generated from the analytical solution of the time-independent Schrödinger equation (TISE), and remains static across all time steps. A key challenge addressed in this study is forecasting the time-dependent behavior of the wavefunction using only these time-invariant, noisy spatial observations. Despite this limitation, the PINN effectively learns the correct temporal evolution by embedding the TDSE into its loss function, combining physics-based residuals with data-fitting terms. The results demonstrate that the model successfully reconstructs both the real and imaginary parts of the wavefunction across different times, accurately denoising the input and preserving quantum dynamics. This study highlights the potential of PINNs in quantum state forecasting, particularly in scenarios where observational data is limited, noisy, or lacks temporal diversity.