# Thermodynamics-informed graph neural networks

arXiv: https://arxiv.org/abs/2203.01874 · HTML: https://mlsanity.com/paper/2203.01874/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 51 references that are in the map.
- First posted on arXiv 3 Mar 2022.

## Builds on (most cited first, 30 of 31)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [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)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [Point-GNN: Graph Neural Network for 3D Object Detection in a Point Cloud](https://mlsanity.com/paper/2003.01251/) (2020)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [GaAN: Gated Attention Networks for Learning on Large and Spatiotemporal Graphs](https://mlsanity.com/paper/1803.07294/) (2018)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Mish: A Self Regularized Non-Monotonic Activation Function](https://mlsanity.com/paper/1908.08681/) (2019)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [ForceNet: A Graph Neural Network for Large-Scale Quantum Calculations](https://mlsanity.com/paper/2103.01436/) (2021)
- [Structure-preserving neural networks](https://mlsanity.com/paper/2004.04653/) (2020)
- [Machine learning structure preserving brackets for forecasting irreversible processes](https://mlsanity.com/paper/2106.12619/) (2021)
- [GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems](https://mlsanity.com/paper/2109.00092/) (2021)
- [Deep learning of thermodynamics-aware reduced-order models from data](https://mlsanity.com/paper/2007.03758/) (2020)
- [ST-UNet: A Spatio-Temporal U-Network for Graph-structured Time Series Modeling](https://mlsanity.com/paper/1903.05631/) (2019)
- [Dual-Primal Graph Convolutional Networks](https://mlsanity.com/paper/1806.00770/) (2018)
- [A Deep Emulator for Secondary Motion of 3D Characters](https://mlsanity.com/paper/2103.01261/) (2021)

## Built on by (most cited first, 30 of 33)

- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [Scalable algorithms for physics-informed neural and graph networks](https://mlsanity.com/paper/2205.08332/) (2022)
- [Gibbs-Duhem-Informed Neural Networks for Binary Activity Coefficient Prediction](https://mlsanity.com/paper/2306.07937/) (2023)
- [Port-metriplectic neural networks: thermodynamics-informed machine learning of complex physical systems](https://mlsanity.com/paper/2211.01873/) (2022)
- [Thermodynamics of learning physical phenomena](https://mlsanity.com/paper/2207.12749/) (2022)
- [GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applications](https://mlsanity.com/paper/2406.03569/) (2024)
- [MeshGraphNet-Transformer: Scalable Mesh-based Learned Simulation for Solid Mechanics](https://mlsanity.com/paper/2601.23177/) (2026)
- [Thermodynamics-informed neural networks for physically realistic mixed reality](https://mlsanity.com/paper/2210.13414/) (2022)
- [Physics-Aware Neural Implicit Solvers for multiscale, parametric PDEs with applications in heterogeneous media](https://mlsanity.com/paper/2405.19019/) (2024)
- [A comparison of Single- and Double-generator formalisms for Thermodynamics-Informed Neural Networks](https://mlsanity.com/paper/2404.01060/) (2024)
- [Symplectic Neural Flows for Modeling and Discovery](https://mlsanity.com/paper/2412.16787/) (2024)
- [On the under-reaching phenomenon in message-passing neural PDE solvers: revisiting the CFL condition](https://mlsanity.com/paper/2507.08861/) (2025)
- [Thermodynamics-informed super-resolution of scarce temporal dynamics data](https://mlsanity.com/paper/2402.17506/) (2024)
- [A hybrid numerical methodology coupling Reduced Order Modeling and Graph Neural Networks for non-parametric geometries: applications to structural dynamics problems](https://mlsanity.com/paper/2406.02615/) (2024)
- [Data-driven particle dynamics: Structure-preserving coarse-graining for emergent behavior in non-equilibrium systems](https://mlsanity.com/paper/2508.12569/) (2025)
- [Physics-Informed Graph Neural Networks to Reconstruct Local Fields Considering Finite Strain Hyperelasticity](https://mlsanity.com/paper/2507.05291/) (2025)
- [Parameter-Efficient Conditioning for Material Generalization in Graph-Based Simulators](https://mlsanity.com/paper/2511.05456/) (2025)
- [Metriplector: From Field Theory to Neural Architecture](https://mlsanity.com/paper/2603.29496/) (2026)
- [Learning Paradigms and Modelling Methodologies for Digital Twins in Process Industry](https://mlsanity.com/paper/2407.02275/) (2024)
- [Nonlinear GENERIC-Embedded Neural Networks (N-GENNs): Learning GENERIC dynamics with non-quadratic dissipation potentials](https://mlsanity.com/paper/2605.09058/) (2026)
- [Critical Sampling for Robust Evolution Operator Learning of Unknown Dynamical Systems](https://mlsanity.com/paper/2304.07485/) (2023)
- [On the Interplay of Subset Selection and Informed Graph Neural Networks](https://mlsanity.com/paper/2306.10066/) (2023)
- [Learning thermodynamic master equations for open quantum systems](https://mlsanity.com/paper/2506.01882/) (2025)
- [Enhancing material behavior discovery using embedding-oriented Physically-Guided Neural Networks with Internal Variables](https://mlsanity.com/paper/2508.00959/) (2025)
- [Constraint Breeds Generalization: Temporal Dynamics as an Inductive Bias](https://mlsanity.com/paper/2512.23916/) (2025)
- [Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs](https://mlsanity.com/paper/2601.11259/) (2026)
- [Graph-Instructed Neural Networks for parametric problems with varying boundary conditions](https://mlsanity.com/paper/2603.08304/) (2026)
- [Variational Graph Neural Networks for Uncertainty Quantification in Inverse Problems](https://mlsanity.com/paper/2603.29515/) (2026)
- [Non-linear mechanical field reconstruction coupling recurrent neural networks with physics-informed graph neural networks](https://mlsanity.com/paper/2606.10909/) (2026)

## Abstract (arXiv)

In this paper we present a deep learning method to predict the temporal evolution of dissipative dynamic systems. We propose using both geometric and thermodynamic inductive biases to improve accuracy and generalization of the resulting integration scheme. The first is achieved with Graph Neural Networks, which induces a non-Euclidean geometrical prior with permutation invariant node and edge update functions. The second bias is forced by learning the GENERIC structure of the problem, an extension of the Hamiltonian formalism, to model more general non-conservative dynamics. Several examples are provided in both Eulerian and Lagrangian description in the context of fluid and solid mechanics respectively, achieving relative mean errors of less than 3% in all the tested examples. Two ablation studies are provided based on recent works in both physics-informed and geometric deep learning.