  We formulate a general framework for hp-variational physics-informed neural
networks (hp-VPINNs) based on the nonlinear approximation of shallow and deep
neural networks and hp-refinement via domain decomposition and projection onto
space of high-order polynomials. The trial space is the space of neural
network, which is defined globally over the whole computational domain, while
the test space contains the piecewise polynomials. Specifically in this study,
the hp-refinement corresponds to a global approximation with local learning
algorithm that can efficiently localize the network parameter optimization. We
demonstrate the advantages of hp-VPINNs in accuracy and training cost for
several numerical examples of function approximation and solving differential
equations.
