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

2025-09-07

AI Summary

Neural operators exhibit poor generalization under low-data regimes and often fail to explicitly encode causality and local temporal structure inherent in physical evolution; autoregressive modeling further exacerbates error accumulation. Method: We propose a forward surrogate model that synergistically integrates a Graph Neural Simulator (GNS) with explicit time-stepping schemes (e.g., Euler integration), directly learning instantaneous time derivatives to solve time-dependent PDEs—thereby explicitly embedding causality and local dynamics. Additionally, we introduce a PCA- and K-means–based trajectory selection strategy to enhance representation efficiency in data-scarce settings. Contribution/Results: On three canonical PDEs, our method achieves <1% relative L² error using only 3% of the training data. In long-term forecasting, it reduces error by 82.48%–99.86% on average over baselines including DeepONet and FNO, significantly advancing low-data PDE surrogate modeling.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Improving data efficiency for neural PDE surrogates with limited training data
• Reducing error accumulation in autoregressive time-dependent PDE modeling
• Encoding causal local-in-time structure in neural operator frameworks

Innovation

• Graph Neural Simulators with explicit time-stepping schemes
• Message-passing GNN framework modeling instantaneous derivatives
• PCA+KMeans trajectory selection for enhanced data efficiency