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

arXiv: https://arxiv.org/abs/2509.06154 · HTML: https://mlsanity.com/paper/2509.06154/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 37 references that are in the map.
- First posted on arXiv 7 Sep 2025.

## Builds on (most cited first, 25 of 25)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Temporal Graph Networks for Deep Learning on Dynamic Graphs](https://mlsanity.com/paper/2006.10637/) (2020)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021)
- [Applications of physics informed neural operators](https://mlsanity.com/paper/2203.12634/) (2022)
- [TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolation](https://mlsanity.com/paper/2505.17341/) (2025)
- [Time Marching Neural Operator FE Coupling: AI Accelerated Physics Modeling](https://mlsanity.com/paper/2504.11383/) (2025)
- [Neural Operators for Stochastic Modeling of Nonlinear Structural System Response to Natural Hazards](https://mlsanity.com/paper/2502.11279/) (2025)
- [Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inference](https://mlsanity.com/paper/2508.05190/) (2025)

## Built on by (most cited first, 2 of 2)

- [ANCHOR: Error-Controlled Adaptive Numerical Correction for Neural Operator Time Marching](https://mlsanity.com/paper/2512.19643/) (2025)
- [AeTHERON: Autoregressive Topology-aware Heterogeneous Graph Operator Network for Fluid-Structure Interaction](https://mlsanity.com/paper/2604.13369/) (2026)

## Abstract (arXiv)

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.