Isogeometric function spaces over parameterized computational domains are generated by the composition of spline basis functions with the inverse of the parameterization. In isogeometric analysis, these spaces are used to represent approximations to functions defined over the computational domain as well as approximations to solutions to partial differential equations. When the function that is to be approximated has a singularity, it is usually necessary to increase the number of degrees of freedom around the singularity, which is typically done by adaptive refinement. In this talk I will present ongoing research on an alternative to adaptive refinement where we do not increase the total number of degrees of freedom. Instead, we optimize the parameterization of the computational domain in such a way that the use of the current degrees of freedom is as effective as possible. To this end, we employ a residual neural network that was trained to predict the optimal parameterization of a three-dimensional point cloud of fixed size for fitting with a quadratic polynomial surface. We evaluate this neural network on a number of point clouds sampled from the computational domain and find a parameterization that best approximates the network’s prediction. In our experimental results, we observe that our method results in an significant improvement of the approximation error compared to classical refinement of the discrete space, also in the multi-patch case.

Joint work with Dany Rios and Thomas Takacs.