  Differentiable programming is a new programming paradigm which enables large
scale optimization through automatic calculation of gradients also known as
auto-differentiation. This concept emerges from deep learning, and has also
been generalized to tensor network optimizations. Here, we extend the
differentiable programming to tensor networks with isometric constraints with
applications to multiscale entanglement renormalization ansatz (MERA) and
tensor network renormalization (TNR). By introducing several gradient-based
optimization methods for the isometric tensor network and comparing with
Evenbly-Vidal method, we show that auto-differentiation has a better
performance for both stability and accuracy. We numerically tested our methods
on 1D critical quantum Ising spin chain and 2D classical Ising model. We
calculate the ground state energy for the 1D quantum model and internal energy
for the classical model, and scaling dimensions of scaling operators and find
they all agree with the theory well.
