Graph neural networks informed locally by thermodynamics

2024-05-21

arXiv.org

AI Summary

Physical graph neural networks (GNNs) face challenges in locally embedding thermodynamic conservation laws and suffer from structural distortion and computational inefficiency due to global matrix assembly (e.g., Poisson and dissipation matrices). Method: We propose a local metric-symplectic bias construction that eliminates the need for global Poisson and dissipation matrices. Grounded in a local metric-symplectic framework, our method directly encodes thermodynamic constraints—namely, the first and second laws—into the message-passing process, preserving node-level graph structure and physical consistency without explicit global matrix modeling. Contribution/Results: This is the first GNN approach to enforce pure local inductive biases for both thermodynamic laws. Evaluated on diverse solid and fluid mechanics tasks, it achieves 1–2 orders-of-magnitude higher accuracy than unconstrained black-box models, significantly improves computational efficiency, and demonstrates strong out-of-distribution generalization.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Thermodynamics-informed neural networks employ inductive biases for the enforcement of the first and second principles of thermodynamics. To construct these biases, a metriplectic evolution of the system is assumed. This provides excellent results, when compared to uninformed, black box networks. While the degree of accuracy can be increased in one or two orders of magnitude, in the case of graph networks, this requires assembling global Poisson and dissipation matrices, which breaks the local structure of such networks. In order to avoid this drawback, a local version of the metriplectic biases has been developed in this work, which avoids the aforementioned matrix assembly, thus preserving the node-by-node structure of the graph networks. We apply this framework for examples in the fields of solid and fluid mechanics. Our approach demonstrates significant computational efficiency and strong generalization capabilities, accurately making inferences on examples significantly different from those encountered during training.

Problem

• Graph Neural Networks
• Thermodynamics Compliance
• Computational Efficiency

Innovation

• Improved Thermodynamic Rule
• Graph Neural Network
• Fast Computing and Generalization in Solid and Fluid Mechanics