Adaptive optimization of isogeometric multi-patch discretizations using artificial neural networks

TL;DR

This paper introduces a neural network-based adaptive method for optimizing isogeometric multi-patch discretizations, improving approximation accuracy for PDE solutions while maintaining tensor-product structure.

Contribution

It presents a novel r-adaptive approach using neural networks to select optimal reparameterizations of multi-patch domains for better PDE approximation.

Findings

Significant reduction in approximation error across tested PDE problems.

Neural network effectively predicts optimal control points for reparameterization.

Method preserves tensor-product structure while enhancing accuracy.

Abstract

In isogeometric analysis, isogeometric function spaces are employed for accurately representing the solution to a partial differential equation (PDE) on a parameterized domain. They are generated from a tensor-product spline space by composing the basis functions with the inverse of the parameterization. Depending on the geometry of the domain and on the data of the PDE, the solution might not have maximum Sobolev regularity, leading to a reduced convergence rate. In this case it is necessary to reduce the local mesh size close to the singularities. The classical approach is to perform adaptive h-refinement, which either leads to an unnecessarily large number of degrees of freedom or to a spline space that does not possess a tensor-product structure. Based on the concept of r-adaptivity we present a novel approach for finding a suitable isogeometric function space for a given PDE without sacrificing the tensor-product structure of the underlying spline space. In particular, we use the fact that different reparameterizations of the same computational domain lead to different isogeometric function spaces while preserving the geometry. Starting from a multi-patch domain consisting of bilinearly parameterized patches, we aim to find the biquadratic multi-patch parameterization that leads to the isogeometric function space with the smallest best approximation error of the solution. In order to estimate the location of the optimal control points, we employ a trained residual neural network that is applied to the graph surfaces of the approximated solution and its derivatives. In our experimental results, we observe that our new method results in a vast improvement of the approximation error for different PDE problems on multi-patch domains.