hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition
This work addresses computational efficiency for researchers solving differential equations, but it is incremental as it builds on existing VPINNs methods.
The authors tackled the challenge of improving accuracy and reducing training costs in physics-informed neural networks by introducing hp-VPINNs, which combine neural networks with domain decomposition and high-order polynomials, resulting in demonstrated efficiency gains in numerical examples.
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.