A Hybrid Semi‐Inverse Variational and Machine Learning Approach for the Schrödinger Equation

ABSTRACT This study introduces a hybrid analytical–machine learning framework for solving the Schrödinger equation with complex potentials. The semi‐inverse variational method is first used to generate highly accurate eigenfunctions and eigenenergies for both 1D radial potentials (Yukawa and Cornell) and a 2D coupled anharmonic oscillator. Based on these rigorous, physics‐consistent results, we train supervised machine learning models; including Random Forest and Neural Network regressors; to predict energy eigenvalues across wide parameter ranges. Both models achieve near‐perfect predictive accuracy (R 2 > 0.999) with errors of only a few millielectronvolts, while preserving fundamental quantum‐mechanical trends. Feature importance analysis confirms that the quantum number n and potential strength parameters dominate the energy scaling, in agreement with theoretical expectations. By integrating variational physics with data‐driven emulation, this hybrid framework reduces computational cost by orders of magnitude; enabling rapid, high‐throughput exploration of quantum systems across dimensions. The approach not only accelerates parameter screening but also serves as a discovery tool, uncovering emergent scaling laws and critical confinement behavior in mixed potentials. This synergy between analytical rigor and machine learning efficiency opens new pathways for quantum simulation, materials design, and the discovery of novel quantum phenomena.