2 research outputs found
Learning the Solution Operator of Boundary Value Problems using Graph Neural Networks
As an alternative to classical numerical solvers for partial differential
equations (PDEs) subject to boundary value constraints, there has been a surge
of interest in investigating neural networks that can solve such problems
efficiently. In this work, we design a general solution operator for two
different time-independent PDEs using graph neural networks (GNNs) and spectral
graph convolutions. We train the networks on simulated data from a finite
elements solver on a variety of shapes and inhomogeneities. In contrast to
previous works, we focus on the ability of the trained operator to generalize
to previously unseen scenarios. Specifically, we test generalization to meshes
with different shapes and superposition of solutions for a different number of
inhomogeneities. We find that training on a diverse dataset with lots of
variation in the finite element meshes is a key ingredient for achieving good
generalization results in all cases. With this, we believe that GNNs can be
used to learn solution operators that generalize over a range of properties and
produce solutions much faster than a generic solver. Our dataset, which we make
publicly available, can be used and extended to verify the robustness of these
models under varying conditions
Towards Learning Self-Organized Criticality of Rydberg Atoms using Graph Neural Networks
Self-Organized Criticality (SOC) is a ubiquitous dynamical phenomenon
believed to be responsible for the emergence of universal scale-invariant
behavior in many, seemingly unrelated systems, such as forest fires, virus
spreading or atomic excitation dynamics. SOC describes the buildup of
large-scale and long-range spatio-temporal correlations as a result of only
local interactions and dissipation. The simulation of SOC dynamics is typically
based on Monte-Carlo (MC) methods, which are however numerically expensive and
do not scale beyond certain system sizes. We investigate the use of Graph
Neural Networks (GNNs) as an effective surrogate model to learn the dynamics
operator for a paradigmatic SOC system, inspired by an experimentally
accessible physics example: driven Rydberg atoms. To this end, we generalize
existing GNN simulation approaches to predict dynamics for the internal state
of the node. We show that we can accurately reproduce the MC dynamics as well
as generalize along the two important axes of particle number and particle
density. This paves the way to model much larger systems beyond the limits of
traditional MC methods. While the exact system is inspired by the dynamics of
Rydberg atoms, the approach is quite general and can readily be applied to
other systems