In finite element analysis, basis functions and integration rules for higher-dimensional spaces are usually constructed by means of tensor product operations over 1D functions and rules. However, resulting spaces are often larger than necessary, leading to a computational overhead that increases with the dimension and the degree of the basis functions. In this work, we present a new method to generate efficient 2D and 3D quadrature rules by leveraging on the concept of trunk spaces [1]. Given the maximum degree $p$ of the basis functions in 1D, we define the trial and test spaces $U$ and $V$ as 2D or 3D trunk spaces of degree $p$. Then, we build our target space as the tensor product $\mathcal{S} = U \otimes V$. To find the quadrature rules for $\mathcal{S}$, we define an optimization problem with parameters $q, X, Y, (Z$ for 3D spaces) and $W$, where $q$ is the number of quadrature points, $X, Y$ and $Z$ are the coordinates of the points, and $W$ the weights. The problem is solved with a single-layer neural network with linear activations using the target loss function defined in [2], and a random restarting strategy to avoid local minima and to minimize $q$. Interestingly, this approach supports the discovery of multiple exact quadrature rules for the same space, which may be useful to address overfitting issues if the rules are used in the training loop of machine learning models. We obtain quadrature rules with an error below $10^{-20}$ for single-element 2D spaces with $p\leq10$ and for 3D spaces with $p\leq6$. The table below shows the savings of these rules in terms of the proportion of reduced integration points with respect to classical Gaussian quadrature rules obtained by tensor product: Savings according to p Dimensionality 1 2 3 4 5 6 7 8 9 10 2D 0% 0% 19% 24% 28% 27% 28% 28% 29% 30% 3D 0% 11% 33% 41% 45% 48% References: [1] A. Düster et al. “The p-version of the Finite Element Method for Three-dimensional Curved Thin Walled Structures”. In: International Journal for Numerical Methods in Engineering 52.7 (Nov. 2001), pp. 673–703. issn: 00295981. doi: 10.1002/nme.222. [2] T. Teijeiro et al. “Machine Learning Discovery of Optimal Quadrature Rules for Isogeometric Analysis”. In: Computer Methods in Applied Mechanics and Engineering 416 (Nov. 2023), p. 116310. issn: 00457825. doi: 10.1016/j.cma.2023.116310.