GNRK: Graph Neural Runge-Kutta method for solving partial differential equations
This provides a more versatile and efficient neural solver for PDEs, addressing limitations in applicability and resolution changes, though it is incremental as it builds on existing neural and classical methods.
The paper tackles the problem of solving partial differential equations (PDEs) with neural networks by introducing GNRK, a method that integrates graph neural networks with a Runge-Kutta structure, achieving superior accuracy and smaller model size compared to existing neural PDE solvers on the 2D Burgers' equation.
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.