Learning Geometric-Aware Quadrature Rules for Functional Minimization

Abstract: Accurate numerical integration over non-uniform point clouds is a challenge for modern mesh-free machine learning solvers for partial differential equations (PDEs) using variational principles. While standard Monte Carlo (MC) methods are not capable of handling a non-uniform point cloud, modern neural network architectures can deal with permutation-invariant inputs, creating quadrature rules for any point cloud. In this work, we introduce QuadrANN, a Graph Neural Network (GNN) architecture designed to learn optimal quadrature weights directly from the underlying geometry of point clouds. The design of the model exploits a deep message-passing scheme where the initial layer encodes rich local geometric features from absolute and relative positions as well as an explicit local density measure. In contrast, the following layers incorporate a global context vector. These architectural choices allow the QuadrANN to generate a data-driven quadrature rule that is permutation-invariant and adaptive to both local point density and the overall domain shape. We test our methodology on a series of challenging test cases, including integration on convex and non-convex domains and estimating the solution of the Heat and Fokker-Planck equations. Across all the tests, QuadrANN reduces the variance of the integral estimation compared to standard Quasi-Monte Carlo methods by warping the point clouds to be more dense in critical areas where the integrands present certain singularities. This enhanced stability in critical areas of the domain at hand is critical for the optimization of energy functionals, leading to improved deep learning-based variational solvers.