Physics-Informed Neural Networks for Quantum Eigenvalue Problems
This work addresses eigenvalue problems in quantum mechanics, offering a data-free approach, but it is incremental as it builds on existing physics-informed neural network methods.
The authors tackled quantum eigenvalue problems by developing an unsupervised neural network method with physics-informed loss functions, achieving analytical and differentiable solutions that satisfy boundary conditions for problems like the finite well and hydrogen atom.
Eigenvalue problems are critical to several fields of science and engineering. We expand on the method of using unsupervised neural networks for discovering eigenfunctions and eigenvalues for differential eigenvalue problems. The obtained solutions are given in an analytical and differentiable form that identically satisfies the desired boundary conditions. The network optimization is data-free and depends solely on the predictions of the neural network. We introduce two physics-informed loss functions. The first, called ortho-loss, motivates the network to discover pair-wise orthogonal eigenfunctions. The second loss term, called norm-loss, requests the discovery of normalized eigenfunctions and is used to avoid trivial solutions. We find that embedding even or odd symmetries to the neural network architecture further improves the convergence for relevant problems. Lastly, a patience condition can be used to automatically recognize eigenfunction solutions. This proposed unsupervised learning method is used to solve the finite well, multiple finite wells, and hydrogen atom eigenvalue quantum problems.