Stochastic Quadrature Rules for Solving PDEs using Neural Networks

We examine the challenges associated with numerical integration when applying Neural Networks to solve Partial Differential Equations (PDEs). We specifically investigate the Deep Ritz Method (DRM), chosen for its practical applicability and known sensitivity to integration inaccuracies. Our research demonstrates that both standard deterministic integration techniques and biased stochastic quadrature methods can lead to incorrect solutions. In contrast, employing high-order, unbiased stochastic quadrature rules defined on integration meshes in low dimensions is shown to significantly enhance convergence rates at a comparable computational expense with respect to low-order methods like Monte Carlo. Additionally, we introduce novel stochastic quadrature approaches designed for triangular and tetrahedral mesh elements, offering increased adaptability for handling complex geometric domains. We highlight that the variance inherent in the stochastic gradient acts as a bottleneck for convergence. Furthermore, we observe that for gradient-based optimisation, the crucial factor is the accurate integration of the gradient, rather than just minimizing the quadrature error of the loss function itself.