Influence of test function order on required quadrature points in VPINN-type methods

Determine how the order of the test functions influences the minimum number of Gauss quadrature points per element required to accurately approximate the weak-form residual integrals in variational physics-informed neural network frameworks, including VPINN, hp-VPINN, and cv-PINN.

Background

Variational PINN frameworks such as VPINN, hp-VPINN, and cv-PINN compute weak-form residuals using Gauss quadrature with test functions (typically Legendre polynomials) that vanish at element boundaries. Prior work has not optimized the pairing of test function order and the number of quadrature points, which affects both computational cost and the fidelity of the weak-form approximation.

The paper highlights that these methods often use many high-order test functions and quadrature points without principled guidance, leading to inefficiencies and a loss of flux information across elements. The explicit open question calls for a clear characterization of how test function order should guide the selection of quadrature points to accurately evaluate the variational residuals.