Graph neural networks informed locally by thermodynamics

TL;DR

This paper introduces a local thermodynamics-informed approach for graph neural networks that maintains their structure, improves computational efficiency, and enhances generalization in solid and fluid mechanics applications.

Contribution

A novel local metriplectic bias for graph neural networks that avoids global matrix assembly, preserving local structure and improving efficiency.

Findings

Significant computational efficiency gains.

Strong generalization to unseen examples.

Accurate inferences in solid and fluid mechanics.

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.