Projection-tree reduced-order modeling for fast N-body computations

This work presents a data-driven reduced-order modeling framework to accelerate the computations of $N$-body dynamical systems and their pair-wise interactions. The proposed differs from traditional acceleration methods, like Barnes-Hut method, which requires online tree building state space, or fast-multipole rigorous $a$ $priori$ analysis governing kernels building. Our approach combines hierarchical decomposition, dimensional compression via least-squares Petrov-Galerkin (LSPG) projection, hyper-reduction by way Gauss-Newton with approximated tensor (GNAT) approach. resulting $projection-tree$ reduced order model (PTROM) enables drastic reduction in operational count complexity constructing sparse hyper-reduced pairwise interactions system. As result, presented is capable achieving an that independent $N$, number bodies numerical domain. Capabilities PTROM method are demonstrated on two-dimensional fluid-dynamic Biot-Savart kernel within parametric reproductive setting. Results show over 2000$\times$ wall-time speed-up respect full-order model, where increases $N$. solution delivers quantities interest errors less than 0.1$\%$ model.