hp-VPINNs: Variational physics-informed neural networks with domain decomposition - Kharazmi, Ehsan; Zhang, Zhongqiang; Karniadakis, George E.M. (Computer methods in applied mechanics and engineering, 2021)
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 the space of high-order polynomials. The trial space is the space of neural network, which is defined globally over the entire computational domain, while the test space contains piecewise polynomials. Specifically in this study, the hp-refinement corresponds to a global approximation with a local learning algorithm that can efficiently localize the network parameter optimization. We demonstrate the advantages of hp-VPINNs in both accuracy and training cost for several numerical examples of function approximation and in solving differential equations. •Development of a general framework for hp-variational physics-informed neural networks•Nonlinear approximation of neural networks, projection onto space of high-order polynomials.•Domain decomposition•Comparison with other methods that use neural networks•Local and global approximations with locally/globally defined test functions.•Different loss functions based on the variational form and integration by parts.•Detailed derivation of the hp-VPINN formulation.