Learning Geometric-Aware Quadrature Rules for Functional Minimization

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.

A deep implicit-explicit minimizing movement method for option pricing in jump-diffusion models

We develop a novel deep learning approach for pricing European basket options written on assets that follow jump-diffusion dynamics. The option pricing problem is formulated as a partial integro-differential equation, which is approximated via a new implicit-explicit minimizing movement time-stepping approach, involving approximation by deep, residual-type Artificial Neural Networks (ANNs) for each time step. The integral operator is discretized via two different approaches: a) a sparse-grid Gauss--Hermite approximation following localised coordinate axes arising from singular value decompositions, and b) an ANN-based high-dimensional special-purpose quadrature rule. Crucially, the proposed ANN is constructed to ensure the asymptotic behavior of the solution for large values of the underlyings and also leads to consistent outputs with respect to a priori known qualitative properties of the solution. The performance and robustness with respect to the dimension of the methods are assessed in a series of numerical experiments involving the Merton jump-diffusion model.