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Graph neural networks informed locally by thermodynamics

Alicia Tierz, Iciar Alfaro, David González, Francisco Chinesta, Elías Cueto

TL;DR

The paper addresses learning dynamic physical systems while strictly enforcing thermodynamic laws. It introduces a local port-metriplectic (GENERIC) framework embedded in graph neural networks to preserve energy and entropy evolution without assembling large global matrices. The approach demonstrates improved accuracy, memory efficiency, and strong generalization across solid and fluid problems, including out-of-distribution geometries and large-scale sloshing simulations. This method offers practical scalability and robustness for physics-informed learning in complex, dissipative environments.

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.

Graph neural networks informed locally by thermodynamics

TL;DR

The paper addresses learning dynamic physical systems while strictly enforcing thermodynamic laws. It introduces a local port-metriplectic (GENERIC) framework embedded in graph neural networks to preserve energy and entropy evolution without assembling large global matrices. The approach demonstrates improved accuracy, memory efficiency, and strong generalization across solid and fluid problems, including out-of-distribution geometries and large-scale sloshing simulations. This method offers practical scalability and robustness for physics-informed learning in complex, dissipative environments.

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.
Paper Structure (17 sections, 32 equations, 10 figures, 3 tables)

This paper contains 17 sections, 32 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methodology
  4. Problem Statement
  5. Geometric structure: Graph Neural Networks
  6. Metriplectic structure: The GENERIC formalism.
  7. Limitations of thermodynamics-informed graph neural networks
  8. A local implementation of thermodynamics-informed GNNs
  9. Port-metriplectic formalism
  10. Learning procedure
  11. Numerical experiments
  12. 3D Viscoelastic Beam Bending
  13. 2D Viscoelastic bending beam
  14. Description of the experiment
  15. Strong out-of-distribution generalization
  16. ...and 2 more sections

Figures (10)

  • Figure 1: Scheme of the discretisation of a physical system by converting a mesh into a graph with which to train our GNN.
  • Figure 2: Scheme of the information processing within a thermodynamics-informed Graph Neural Network.
  • Figure 3: Sketch of the viscous-hyperelastic 3D beam bending problem.
  • Figure 4: Zero-shot performance for 4 different trajectories. RMSE and RRMSE values for position ($\boldsymbol{q}$), velocity $\dot{\boldsymbol{q}}$ and von Mises stress ($\boldsymbol{\sigma}$). Each point on the plot represents the error an individual trajectory. The blue boxes represent the error distribution for the SPNN, the light grey boxes show the distribution for the vanilla-GNNS, the burgundy boxes show the error distribution for the global TIGNN and, finally, the yellow boxes represent the results for the just proposed technique.
  • Figure 5: The results depict the predictions for the three test simulations. Figures (a), (b), and (c) illustrate the network's inferred outputs at sample time instants, while figures (d), (e), and (f) depict the ground truth for comparison. The color coding denotes the $xx$ component of the stress tensor.
  • ...and 5 more figures