GNRK: Graph Neural Runge-Kutta method for solving partial differential
  equations
Hoyun Choi Sungyeop Lee B. Kahng Junghyo Jo
Neural networks have proven to be efficient surrogate models for tackling
partial differential equations (PDEs). However, their applicability is often
confined to specific PDEs under certain constraints, in contrast to classical
PDE solvers that rely on numerical differentiation. Striking a balance between
efficiency and versatility, this study introduces a novel approach called Graph
Neural Runge-Kutta (GNRK), which integrates graph neural network modules with a
recurrent structure inspired by the classical solvers. The GNRK operates on
graph structures, ensuring its resilience to changes in spatial and temporal
resolutions during domain discretization. Moreover, it demonstrates the
capability to address general PDEs, irrespective of initial conditions or PDE
coefficients. To assess its performance, we benchmark the GNRK against existing
neural network based PDE solvers using the 2-dimensional Burgers' equation,
revealing the GNRK's superiority in terms of model size and accuracy.
Additionally, this graph-based methodology offers a straightforward extension
for solving coupled differential equations, typically necessitating more
intricate models.