# Port-metriplectic neural networks: thermodynamics-informed machine learning of complex physical systems

arXiv: https://arxiv.org/abs/2211.01873 · HTML: https://mlsanity.com/paper/2211.01873/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 66 references that are in the map.
- First posted on arXiv 3 Nov 2022.

## Builds on (most cited first, 28 of 28)

- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [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)
- [Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Data](https://mlsanity.com/paper/2012.02334/) (2020)
- [Learning Physical Dynamics with Subequivariant Graph Neural Networks](https://mlsanity.com/paper/2210.06876/) (2022)
- [Unsupervised Learning of Lagrangian Dynamics from Images for Prediction and Control](https://mlsanity.com/paper/2007.01926/) (2020)
- [Deep learning of thermodynamics-aware reduced-order models from data](https://mlsanity.com/paper/2007.03758/) (2020)
- [Physical Design using Differentiable Learned Simulators](https://mlsanity.com/paper/2202.00728/) (2022)
- [Thermodynamics-informed graph neural networks](https://mlsanity.com/paper/2203.01874/) (2022)
- [Deconstructing the Inductive Biases of Hamiltonian Neural Networks](https://mlsanity.com/paper/2202.04836/) (2022)
- [LagNetViP: A Lagrangian Neural Network for Video Prediction](https://mlsanity.com/paper/2010.12932/) (2020)
- [Machine Learning and Computational Mathematics](https://mlsanity.com/paper/2009.14596/) (2020)
- [Pseudo-Hamiltonian Neural Networks with State-Dependent External Forces](https://mlsanity.com/paper/2206.02660/) (2022)
- [Learning ground states of quantum Hamiltonians with graph networks](https://mlsanity.com/paper/2110.06390/) (2021)
- [Physics perception in sloshing scenes with guaranteed thermodynamic consistency](https://mlsanity.com/paper/2106.13301/) (2021)
- [Thermodynamics of learning physical phenomena](https://mlsanity.com/paper/2207.12749/) (2022)
- [A Thermodynamics-informed Active Learning Approach to Perception and Reasoning about Fluids](https://mlsanity.com/paper/2203.05775/) (2022)
- [Learning Neural Hamiltonian Dynamics: A Methodological Overview](https://mlsanity.com/paper/2203.00128/) (2022)
- [Robust port-Hamiltonian representations of passive systems](https://mlsanity.com/paper/1801.05018/) (2018)
- [A unified framework for Hamiltonian deep neural networks](https://mlsanity.com/paper/2104.13166/) (2021)
- [Lagrangian Neural Network with Differentiable Symmetries and Relational Inductive Bias](https://mlsanity.com/paper/2110.03266/) (2021)
- [Machine Learning a Molecular Hamiltonian for Predicting Electron Dynamics](https://mlsanity.com/paper/2007.09814/) (2020)
- [Mastering high-dimensional dynamics with Hamiltonian neural networks](https://mlsanity.com/paper/2008.04214/) (2020)

## Built on by (most cited first, 11 of 11)

- [Pseudo-Hamiltonian neural networks for learning partial differential equations](https://mlsanity.com/paper/2304.14374/) (2023)
- [MeshGraphNet-Transformer: Scalable Mesh-based Learned Simulation for Solid Mechanics](https://mlsanity.com/paper/2601.23177/) (2026)
- [On the under-reaching phenomenon in message-passing neural PDE solvers: revisiting the CFL condition](https://mlsanity.com/paper/2507.08861/) (2025)
- [Metriplector: From Field Theory to Neural Architecture](https://mlsanity.com/paper/2603.29496/) (2026)
- [Physics informed cell representations for variational formulation of multiscale problems](https://mlsanity.com/paper/2405.16770/) (2024)
- [Thermodynamics-informed graph neural networks for real-time simulation of digital human twins](https://mlsanity.com/paper/2412.12034/) (2024)
- [Nonlinear GENERIC-Embedded Neural Networks (N-GENNs): Learning GENERIC dynamics with non-quadratic dissipation potentials](https://mlsanity.com/paper/2605.09058/) (2026)
- [SALSA-RL: Stability Analysis in the Latent Space of Actions for Reinforcement Learning](https://mlsanity.com/paper/2502.15512/) (2025)
- [Learning thermodynamic master equations for open quantum systems](https://mlsanity.com/paper/2506.01882/) (2025)
- [A Graph Neural Network approach to zero-shot Digital Twins](https://mlsanity.com/paper/2607.20535/) (2026)
- [An Energy-Based Conservative-Dissipative Latent Neural Evolution Operator for Magnetization Dynamics](https://mlsanity.com/paper/2609.04530/) (2026)

## Abstract (arXiv)

We develop inductive biases for the machine learning of complex physical systems based on the port-Hamiltonian formalism. To satisfy by construction the principles of thermodynamics in the learned physics (conservation of energy, non-negative entropy production), we modify accordingly the port-Hamiltonian formalism so as to achieve a port-metriplectic one. We show that the constructed networks are able to learn the physics of complex systems by parts, thus alleviating the burden associated to the experimental characterization and posterior learning process of this kind of systems. Predictions can be done, however, at the scale of the complete system. Examples are shown on the performance of the proposed technique.