  Deep neural network (DNN) and auto differentiation have been widely used in
computational physics to solve variational problems. When DNN is used to
represent the wave function to solve quantum many-body problems using
variational optimization, various physical constraints have to be injected into
the neural network by construction, to increase the data and learning
efficiency. We build the unitary constraint to the variational wave function
using a monotonic neural network to represent the Cumulative Distribution
Function (CDF) $F(x) = \int_{-\infty}^{x} \psi^*\psi dx'$. Using this
constrained neural network to represent the variational wave function, we solve
Schrodinger equations using auto-differentiation and stochastic gradient
descent (SGD), by minimizing the violation of the trial wave function $\psi(x)$
to the Schrodinger equation. For several classical problems in quantum
mechanics, we obtain their ground state wave function and energy with very low
errors. The method developed in the present paper may pave a new way in solving
nuclear many body problems in the future.
