Data-Efficient Time-Dependent PDE Surrogates: Graph Neural Simulators vs Neural Operators

Abstract: Neural operators (NOs) approximate mappings between infinite-dimensional function spaces but require large datasets and struggle with scarce training data. Many NO formulations don't explicitly encode causal, local-in-time structure of physical evolution. While autoregressive models preserve causality by predicting next time-steps, they suffer from rapid error accumulation. We employ Graph Neural Simulators (GNS) - a message-passing graph neural network framework - with explicit numerical time-stepping schemes to construct accurate forward models that learn PDE solutions by modeling instantaneous time derivatives. We evaluate our framework on three canonical PDE systems: (1) 2D Burgers' scalar equation, (2) 2D coupled Burgers' vector equation, and (3) 2D Allen-Cahn equation. Rigorous evaluations demonstrate GNS significantly improves data efficiency, achieving higher generalization accuracy with substantially fewer training trajectories compared to neural operator baselines like DeepONet and FNO. GNS consistently achieves under 1% relative L2 errors with only 30 training samples out of 1000 (3% of available data) across all three PDE systems. It substantially reduces error accumulation over extended temporal horizons: averaged across all cases, GNS reduces autoregressive error by 82.48% relative to FNO AR and 99.86% relative to DON AR. We introduce a PCA+KMeans trajectory selection strategy enhancing low-data performance. Results indicate combining graph-based local inductive biases with conventional time integrators yields accurate, physically consistent, and scalable surrogate models for time-dependent PDEs.