  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.
