Energy-Embedded Neural Solvers for One-Dimensional Quantum Systems

Physics-informed neural networks (PINN) have been widely used in computational physics to solve partial differential equations (PDEs). In this study, we propose an energy-embedding-based physics-informed neural network method for solving the one-dimensional time-independent Schr\"{o}dinger equation to obtain ground- and excited-state wave functions, as well as energy eigenvalues by incorporating an embedding layer to generate process-driven data. The method demonstrates high accuracy for several well-known potentials, such as the infinite potential well, harmonic oscillator potential, Woods-Saxon potential, and double-well potential. Further validation shows that the method also performs well in solving the radial Coulomb potential equation, showcasing its adaptability and extensibility. The proposed approach can be extended to solve other partial differential equations beyond the Schr\"{o}dinger equation and holds promise for applications in high-dimensional quantum systems.