Solving the Schrödinger equation using physics informed neural networks

This thesis seeks to investigate the use of physics‑informed neural networks (PINNs) to obtain ground‑ and excited‑state solutions of the time‑independent Schrödinger equation in the one dimensional infinte square well potential and the hydrogen atom. Two fully connected PINN architectures are compared, one treats the energy E as a learnable parameter, while the other calculates E directly from the network’s prediction via the Hamiltonian residual. The solutions are found through minimization of an appropriate loss function with embedded quantum mechanical principles such as the differential equation residual, normalization condition, boundary values and orthogonality. Results highlight the PINNs ability to solve for the first three energy levels for the one dimensional problem with relative errors under 3.4*10^-3, while wave‑function fits stay below a mean‑squared error of 10^-3 inside the well within 2000 epochs. However, manually tuning of the model proved crucial for achieving accurate results, which makes the method used in this thesis rather hard to automate. Multiple challenges were encountered during the project, especially concerning the hydrogen atom and attempts to avoid local minima in the loss function environment. As described in the discussion, the hydrogen atom seems to fail to converge to reasonable solutions, while the local minima problem could be avoided by introducing special weighting for each loss term. In summary, while implementations are challenging and highly variable with model optimization, using PINNs to solve quantum mechanical problems prove to be promising as results yield accurate solutions to some of the problems, giving the wave functions with their corresponding energies. However, further work is required to solve multi-dimensional problems.