Solving the Schrödinger Equation with Artificial Neural Networks

We aim to combine traditional numerical methods with advanced machine learning algorithms to solve one of quantum mechanics’ fundamental equations: the time-independent Schrödinger equation providing eigenenergies and eigenstates of quantum systems.

Typically, numerical methods, such as finite difference, are used to discretize the spatial domain, converting the differential equation into a matrix eigenvalue problem. We use the eigenenergies obtained from such conventional numerical methods as training data to teach a machine learning algorithm to predict eigenenergies of previously unseen quantum systems. In this context, we use artificial neural networks as universal function approximators that learn the intricate mapping between different potential configurations and their corresponding energy levels. The trained models can replicate the finite difference results and further generalize beyond the specific training cases. This capability is particularly useful when dealing with complex potentials or systems involving multiple particles, where traditional methods become computationally expensive.