Operating system

Algorithm

Physics

Mathematical optimization

Mathematical analysis

Finite element method

Artificial intelligence

Nonlinear system

Piecewise

Domain decomposition methods

Space (punctuation)

Applied mathematics

Mathematics

Domain (mathematical analysis)

Computer science

Artificial neural network

Projection (relational algebra)

Quantum mechanics

Function approximation

Thermodynamics

Computer Methods in Applied Mechanics and Engineering

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.

hp-VPINNs: Variational physics-informed neural networks with domain decomposition