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This research introduces Graph Neural Simulators (GNS) as a superior alternative to standard neural operators for solving time-dependent partial differential equations in data-scarce regimes. By integrating message-passing mechanisms with numerical time-stepping schemes, GNS achieves significantly higher data efficiency and stability compared to existing architectures like DeepONet or FNO. The study demonstrates that leveraging local graph structures and solver-inspired designs is critical for advancing AI for Science applications involving complex physical systems.
Data-Efficient Time-Dependent PDE Surrogates: Graph Neural Simulators vs. Neural Operators
arXiv:2509.06154v3 Announce Type: replace Abstract: Developing accurate, data-efficient surrogate models is central to advancing AI for Science. Neural operators (NOs), which approximate mappings between infinite-dimensional function spaces using conventional neural architectures, have gained popularity as surrogates for systems driven by partial differential equations (PDEs). However, their reliance on large datasets and limited ability to generalize in low-data regimes hinder their practical utility. We argue that these limitations arise from their global processing of data, which fails to exploit the local, discretized structure of physical systems. To address this, we propose Graph Neural Simulators (GNS) as a principled surrogate modeling paradigm for time-dependent PDEs. GNS leverages message-passing combined with numerical time-stepping schemes to learn PDE dynamics by modeling the instantaneous time derivatives. This design mimics traditional numerical solvers, enabling stable long-horizon rollouts and strong inductive biases that enhance generalization. We rigorously evaluate GNS on five canonical PDE systems: (1) 2D scalar Burgers', (2) 2D coupled Burgers', (3) 2D Allen-Cahn, (4) 2D nonlinear shallow water equations, and (5) 2D flow past a cylinder, comparing against SOTA NOs including DeepONet, FNO, and GINO. Results demonstrate that GNS is markedly more data-efficient, achieving less than 1% relative L2 error using only 3% of available trajectories, and exhibits dramatically reduced error accumulation over time (82.5% lower autoregressive error than FNO, 99.9% lower than DeepONet). To choose the training data, we introduce a PCA combined with KMeans trajectory selection strategy. These findings provide compelling evidence that GNS, with its graph-based locality and solver-inspired design, is the most suitable and scalable surrogate modeling framework for AI-driven scientific discovery.
Title:Data-Efficient Time-Dependent PDE Surrogates: Graph Neural Simulators vs. Neural Operators
Abstract:Developing accurate, data-efficient surrogate models is central to advancing AI for Science. Neural operators (NOs), which approximate mappings between infinite-dimensional function spaces using conventional neural architectures, have gained popularity as surrogates for systems driven by partial differential equations (PDEs). However, their reliance on large datasets and limited ability to generalize in low-data regimes hinder their practical utility. We argue that these limitations arise from their global processing of data, which fails to exploit the local, discretized structure of physical systems. To address this, we propose Graph Neural Simulators (GNS) as a principled surrogate modeling paradigm for time-dependent PDEs. GNS leverages message-passing combined with numerical time-stepping schemes to learn PDE dynamics by modeling the instantaneous time derivatives. This design mimics traditional numerical solvers, enabling stable long-horizon rollouts and strong inductive biases that enhance generalization. We rigorously evaluate GNS on five canonical PDE systems: (1) 2D scalar Burgers', (2) 2D coupled Burgers', (3) 2D Allen-Cahn, (4) 2D nonlinear shallow water equations, and (5) 2D flow past a cylinder, comparing against SOTA NOs including DeepONet, FNO, and GINO. Results demonstrate that GNS is markedly more data-efficient, achieving less than 1% relative L2 error using only 3% of available trajectories, and exhibits dramatically reduced error accumulation over time (82.5% lower autoregressive error than FNO, 99.9% lower than DeepONet). To choose the training data, we introduce a PCA combined with KMeans trajectory selection strategy. These findings provide compelling evidence that GNS, with its graph-based locality and solver-inspired design, is the most suitable and scalable surrogate modeling framework for AI-driven scientific discovery.
37 pages including references. Supplementary Information provided
Machine Learning (cs.LG); Computation (stat.CO); Machine Learning (stat.ML)
arXiv:2509.06154 [cs.LG]
(or arXiv:2509.06154v3 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2509.06154 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 370 words
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