Solution of Schrödinger Equation for Quantum Systems via Physics-Informed Neural Networks

The numerous successes achieved by machine learning techniques in many technical areas have sparked interest in the scientific community for their application in science. By merging the knowledge of machine learning experts and computational scientists, the field of scientific machine learning has shown its ability to greatly improve the performance of existing computational methods. One possible approach to developing physics-aware machine learning is the inclusion of physical constraints in the training of a machine learning model. Physics-Informed Neural Networks are an example of such an approach, as they can incorporate prior physical knowledge into their architecture, enabling them to learn and simulate complex phenomena while respecting the underlying physics principles. Possible constraints are physical laws, symmetries, and conservation laws. Compared to other machine learning models, Physics-Informed Neural Networks do not require substantial input data, with the exception of initial and boundary conditions to correctly formalize the problem. In this Thesis, we exploit the advantages of Physics-Informed Neural Networks to efficiently simulate one-electron quantum systems. The simulations rely on the direct solution of the eigenvalue equation represented by the Schrödinger equation. Traditional methods for solving the Schrödinger equation often rely on approximations and can become computationally expensive for nontrivial systems. The mesh-free Physics-Informed Neural Networks approach avoids the need for discretization, as the residuals computed with respect to the physical constraints are minimized during training for a given set of points within the domain. The solution of the Schrödinger equation allows one to calculate important physical quantities of the physical system under study, such as the ground state energy, the electronic wavefunction, and the associated electron density. These quantities are compared with the estimations present in the quantum chemistry literature to assess the performance of the Physics-Informed Machine Learning approach.