Neighbor search in latent spaces via geometric deep learning for nonlocal methods in fluid dynamics*

Nonlocal (NL) numerical methods are algorithmically versatile and enable effective simulations of complex physical phenomena, such as free-surface and multi-phase fluid flows. In contrast to traditional methods that employ local operations, such as finite elements or finite volumes, NL methods rely on NL collocation support to model dynamical systems. It follows that NL methods suffer generally from a lack of sparsity, and incur a higher computational cost than local methods due to the requirement of a NL neighbor search. Recent efforts in NL projection-based model order reduction have attempted to ameliorate this cost bottleneck using dimensional reduction, hyper-reduction, and hierarchical agglomeration. Unfortunately, most of this work relies on hierarchical agglomeration of neighbors in a linear subspace and cannot account for neighbor shifting and evolution. Toward addressing these limitations, the current work aims to leverage graph neural networks, a framework within geometric deep learning, to serve as a time-adaptive NL neighbor search algorithm in a nonlinear latent space. This approach will be applied to the NL smoothed-particle hydrodynamics framework, where case studies will include natural convection instabilities.