A Study on the Stability Limits of Graph Neural Network Surrogates for Advection-Diffusion
M. Maassen van den Brink (TU Delft - Electrical Engineering, Mathematics and Computer Science)
A. Heinlein – Mentor (TU Delft - Electrical Engineering, Mathematics and Computer Science)
R.P. Dwight – Mentor (TU Delft - Aerospace Engineering)
M. Verlaan – Graduation committee member (TU Delft - Electrical Engineering, Mathematics and Computer Science)
More Info
expand_more
Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.
Abstract
Machine-learned surrogate time integrators promise large speed-ups over classical solvers, yet their performance is usually reported as a single aggregate error, leaving open the question of when they remain stable. This thesis determines the empirical stability limits of graph neural network (GNN) surrogates for the two-dimensional advection–diffusion equation on unstructured meshes with periodic boundary conditions, expressed directly in the dimensionless CFL and Fourier numbers.
Surrogates are trained to minimise the one-step error on fixed velocity and diffusion fields and evaluated autoregressively on unseen fields over horizons eight times the training window. Two families are compared at a fixed message passing budget: single-scale models, and multiscale models organised as V-cycles over predetermined coarsened graphs. For each model, a piecewise-linear fit of the final rollout error against the CFL and Fourier numbers yields empirical stability limits, defined by a blow-up threshold.
Within these limits the surrogates reproduce the finite element reference accurately on both seen and unseen fields and show no abrupt change beyond the training horizon, although the diffusion-dominated regime is consistently harder than the advection-dominated one. The single-scale CFL limit tracks the number of message-passing blocks and lies slightly above it. Adding coarse levels at a fixed total message passing layer budget broadens the advective stability range, decisively at the largest stride, but a two-level hierarchy trades diffusive stability and in-region accuracy for this gain. Only a three-level V-cycle removes the penalty, attaining zero blow-ups on both axes, and deeper models show no oversmoothing.
The diffusion-side limits carry large variance, traced partly to the dissipative backward-Euler reference, and should be read as indicative. The work delivers a concrete operational range for GNN surrogates and identifies how multiscale models can extend it.