RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domains

TL;DR

RIGNO introduces a region-interacting graph neural operator for learning PDE solution operators on arbitrary domains. It uses a multi-scale regional mesh with encoder–processor–decoder GNNs, along with edge masking and temporal conditioning, to achieve strong spatial-temporal resolution invariance and accurate, robust predictions across unstructured point clouds and Cartesian grids. The method delivers state-of-the-art results across 13 PDE datasets, demonstrates scalable performance with data and model size, and provides mechanisms for uncertainty quantification and transfer learning. This framework offers a practical, scalable surrogate for complex PDE solvers with broad applicability to engineering problems on irregular geometries.

Abstract

Learning the solution operators of PDEs on arbitrary domains is challenging due to the diversity of possible domain shapes, in addition to the often intricate underlying physics. We propose an end-to-end graph neural network (GNN) based neural operator to learn PDE solution operators from data on point clouds in arbitrary domains. Our multi-scale model maps data between input/output point clouds by passing it through a downsampled regional mesh. The approach includes novel elements aimed at ensuring spatio-temporal resolution invariance. Our model, termed RIGNO, is tested on a challenging suite of benchmarks composed of various time-dependent and steady PDEs defined on a diverse set of domains. We demonstrate that RIGNO is significantly more accurate than neural operator baselines and robustly generalizes to unseen resolutions both in space and in time. Our code is publicly available at github.com/camlab-ethz/rigno.

Figures (53)

• Figure 1: General schematic of the RIGNO architecture for an idealized two-dimensional domain. The inputs are first independently projected to a latent space by feed-forward blocks. The information on the original discretization (physical nodes) is then locally aggregated to a coarser discretization (regional nodes). Regional nodes are connected to each other by edges with multiple length scales. Several message-passing steps are then applied on the regional nodes which constitute the processor. The processed features are then transmitted back to the original discretization by using similar edges as in the encoder, before being independently projected back to the desired output dimension via a feed-forward block without normalization layers.
• Figure 2: Scaling behavior of RIGNO with respect to dataset and model size. All values correspond to autoregressive (AR-2, see SM Section \ref{['app:autoregressive-inference']}) test errors at $t=t_{14}$.
• Figure 3: Ablation of the proposed training strategies. a) The effect of edge masking on resolution invariance. EM stands for edge masking. The y-axis shows the relative test error at $t=t_{14}$ when the spatial resolution of the inputs and outputs varies from the training resolution $64^2$. b) Autoregressive test errors with and without fractional pairing fine-tuning. The initial solution (model input) is updated on vertical lines according to the AR-4 scheme (see SM Section \ref{['app:autoregressive-inference']}). These models are exceptionally trained on snapshots with time resolution $4\Delta t$ and up to time $t_{16}$. c) Autoregressive (AR-2) test errors with different time marching strategies. Time snapshots after the vertical line has not been seen during training and are considered as extrapolation in time.
• Figure A.1: Construction of regional mesh edges and support sub-regions with periodic boundary conditions. The dashed lines show the domain boundaries. Large and small purple dots represent the main regional nodes and the ghost regional nodes, respectively. Black lines represent the main edges, and the purple lines the cross-boundary edges. Note that in Figure \ref{['fig:rmesh-subregions']}, the parts of the domain that are close to the domain boundaries and do not lie in any circle are covered by at least one regional node on the other side of the boundary.
• Figure A.2: Graph structure for periodic boundary conditions in one-dimensional settings. The green dots represent physical nodes, the purple circles represent regional nodes, and the solid lines represent edge connections. Note that only a few physical nodes are visualized. The bottom right part of the figure partly shows the receptive fields of two regional nodes in the encoder. Here, the red lines show the edges that are randomly masked, and hence will not be used in that particular message passing block.