Learning Geometric-Aware Quadrature Rules for Functional Minimization

TL;DR

The paper tackles the challenge of numerical integration on non-uniform, unstructured point clouds within mesh-free variational solvers for PDEs by introducing QuadrANN, a graph neural network that learns quadrature weights directly from geometry. The model combines a density-aware geometric encoding with globally-aware propagation to produce permutation-invariant weights that sum to one, trained on a basis of multivariate polynomials up to degree $J$ with randomized trigonometric augmentation, enforcing exactness for constants via a loss on $\mathcal{C}(\mathcal{X};\theta)$. Empirical results show reduced variance in integral estimates compared to Sobol'-QMC across 2D and 4D tests and improved PDE solutions for the Heat and Fokker-Planck equations, with modest computational overhead relative to standard methods. The work provides a stable, data-driven quadrature framework for energy minimization in mesh-free variational solvers and suggests future directions such as learning the warping map to further optimize point distributions. Key formulas include $\mathcal{J}[u] = \int_\Omega L(x,u,\nabla u)dx$ and the weight-approximation objective $\mathcal{C}(\mathcal{X};\theta) = \sum_{\boldsymbol j} \big(\sum_{i=1}^n W_i(\mathcal{X};\theta) \Phi_{\boldsymbol j}(x_i) - 1\big)^2$, with constants enforced by the softmax constraint on $W_i$.

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.

Figures (5)

• Figure 1: Interpolated quadrature weights learned by QuadrANN for the 2D unit square example. The color map indicates the weight value, and the dots refer to the locations of the points.
• Figure 2: Visualization of the quadrature weights for the L-shaped domain. The model accurately assigns the weight of the points to adapt to the non-convex geometry. The color map indicates the weight value, and the dots refer to the locations of the points.
• Figure 3: Comparison of solutions for the 2D Heat Equation. A 1D slice of the computed solution $u(T,\mathbf{x})$ at the final time $T=1$, taken along the $x_1$-axis at $x_2=0.5$. The plot compares the solution obtained using our proposed QuadrANN integration with the standard Sobol'-based Monte Carlo (MC) method against the Finite Difference Method (FDM) benchmark. The QuadrANN-based solution shows a closer agreement with the benchmark.
• Figure 4: Absolute error at final time $T=1$ relative to the Finite Difference benchmark. The plots compare the error of the solution obtained with QuadrANN integration versus the corresponding Sobol'-QMC solution.
• Figure 5: Comparison of solutions for the 4D Fokker-Planck equation. A 1D slice of the computed density $u(T,\mathbf{x})$ at the final time $T=1/2$, taken along the $x_1$-axis with other coordinates fixed at the center of the domain ($x_2=x_3=x_4=0.5$). The plot compares the density profile from our QuadrANN integration scheme with a standard Sobol'-based Monte Carlo (quasi-MC) method against the benchmark density constructed via SDE path simulation and Kernel Density Estimation (KDE).