Efficient Numerical Integration for Finite Element Trunk Spaces in 2D and 3D using Machine Learning: A new Optimisation Paradigm to Construct Application-Specific Quadrature Rules

Finite element methods usually construct basis functions and quadrature rules for multidimensional domains via tensor products of one-dimensional counterparts. While straightforward, this approach results in integration spaces larger than necessary, especially as the polynomial degree $p$ or the spatial dimension increases, leading to considerable computational overhead. This work starts from the hypothesis that reducing the dimensionality of the polynomial space can lead to quadrature rules with fewer points and lower computational cost, while preserving the exactness of numerical integration. We use trunk spaces that exclude high-degree monomials that do not improve the approximation quality of the discrete space. These reduced spaces retain sufficient expressive power and allow us to construct smaller (more economical) integration domains. Given a maximum degree $p$, we define trial and test spaces $U$ and $V$ as 2D or 3D trunk spaces and form the integration space $\mathcal{S} = U \otimes V$. We then construct exact quadrature rules by solving a non-convex optimisation problem over the number of points $q$, their coordinates, and weights. We use a shallow neural network with linear activations to parametrise the rule, and a random restart strategy to mitigate convergence to poor local minima. When necessary, we dynamically increase $q$ to achieve exact integration. Our construction reaches machine-precision accuracy (errors below 1e-22) using significantly fewer points than standard tensor-product Gaussian quadrature: up to 30\% reduction in 2D for $p \leq 10$, and 50\% in 3D for $p \leq 6$. These results show that combining the mathematical understanding of polynomial structure with numerical optimisation can lead to a practical and extensible methodology for improving the adaptiveness, efficiency, and scalability of quadrature rules for high-order finite element simulations.