  Iterative solvers are widely used to accurately simulate physical systems.
These solvers require initial guesses to generate a sequence of improving
approximate solutions. In this contribution, we introduce a novel method to
accelerate iterative solvers for physical systems with graph networks (GNs) by
predicting the initial guesses to reduce the number of iterations. Unlike
existing methods that aim to learn physical systems in an end-to-end manner,
our approach guarantees long-term stability and therefore leads to more
accurate solutions. Furthermore, our method improves the run time performance
of traditional iterative solvers. To explore our method we make use of
position-based dynamics (PBD) as a common solver for physical systems and
evaluate it by simulating the dynamics of elastic rods. Our approach is able to
generalize across different initial conditions, discretizations, and realistic
material properties. Finally, we demonstrate that our method also performs well
when taking discontinuous effects into account such as collisions between
individual rods. Finally, to illustrate the scalability of our approach, we
simulate complex 3D tree models composed of over a thousand individual branch
segments swaying in wind fields. A video showing dynamic results of our graph
learning assisted simulations of elastic rods can be found on the project
website available at
http://computationalsciences.org/publications/shao-2021-physical-systems-graph-learning.html .
