APPLICATION OF PHYSICS-INFORMED NEURAL NETWORKS (PINNS) IN SOLVING QUANTUM MECHANICS PROBLEMS WITH MULTI-LEVEL COMPLEX BOUNDARIES

This paper investigates the application of physics-informed neural networks (PINNs) to eigenvalue problems governed by partial differential equations, with the two-dimensional time-independent Schrödinger equation taken as a representative example. To address the limitations of traditional numerical methods in handling complex boundaries and high-dimensional problems, we propose a PINN framework in which the energy eigenvalue E is treated as a trainable parameter, together with a hybrid optimization strategy combining Adam and L-BFGS. To compute the ground state and multiple excited states simultaneously, normalization and orthogonality constraints between wave functions are incorporated into the loss function. Numerical experiments are conducted on several geometric domains, including circular, square, pentagram-shaped, and stadium-shaped boundaries. The results show that, for regular and symmetric domains, the PINN method achieves higher accuracy than the finite difference method in calculating low-energy levels. By using independent parameterization and orthogonality constraints, the method can effectively identify energy degeneracies caused by geometric symmetry. For irregular boundaries, the mesh-free nature of PINNs avoids the difficulty of complex mesh generation and demonstrates good adaptability and potential. Owing to its mesh-free formulation, ability to preserve geometric symmetry, and potential for extension to high-dimensional problems, the PINN method provides an effective and promising approach for solving eigenvalue problems with complex boundary conditions.