# Machine learning structure preserving brackets for forecasting irreversible processes

arXiv: https://arxiv.org/abs/2106.12619 · HTML: https://mlsanity.com/paper/2106.12619/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 60 references that are in the map.
- First posted on arXiv 23 Jun 2021.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs](https://mlsanity.com/paper/1902.10298/) (2019)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [Structure-preserving neural networks](https://mlsanity.com/paper/2004.04653/) (2020)
- [Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems](https://mlsanity.com/paper/2010.14685/) (2020)
- [MALI: A memory efficient and reverse accurate integrator for Neural ODEs](https://mlsanity.com/paper/2102.04668/) (2021)
- [Achieving Conservation of Energy in Neural Network Emulators for Climate Modeling](https://mlsanity.com/paper/1906.06622/) (2019)
- [SNODE: Spectral Discretization of Neural ODEs for System Identification](https://mlsanity.com/paper/1906.07038/) (2019)
- [Symplectic Neural Networks in Taylor Series Form for Hamiltonian Systems](https://mlsanity.com/paper/2005.04986/) (2020)
- [Time-Reversal Symmetric ODE Network](https://mlsanity.com/paper/2007.11362/) (2020)
- [ANODEV2: A Coupled Neural ODE Evolution Framework](https://mlsanity.com/paper/1906.04596/) (2019)
- [A Kinetic Monte Carlo Approach for Simulating Cascading Transmission Line Failure](https://mlsanity.com/paper/1912.08081/) (2019)

## Built on by (most cited first, 30 of 52)

- [GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems](https://mlsanity.com/paper/2109.00092/) (2021)
- [One-Shot Transfer Learning of Physics-Informed Neural Networks](https://mlsanity.com/paper/2110.11286/) (2021)
- [Parameterized Physics-informed Neural Networks for Parameterized PDEs](https://mlsanity.com/paper/2408.09446/) (2024)
- [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)
- [Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling](https://mlsanity.com/paper/2109.05364/) (2021)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [tLaSDI: Thermodynamics-informed latent space dynamics identification](https://mlsanity.com/paper/2403.05848/) (2024)
- [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024)
- [Scalable algorithms for physics-informed neural and graph networks](https://mlsanity.com/paper/2205.08332/) (2022)
- [Efficiently Parameterized Neural Metriplectic Systems](https://mlsanity.com/paper/2405.16305/) (2024)
- [Exact conservation laws for neural network integrators of dynamical systems](https://mlsanity.com/paper/2209.11661/) (2022)
- [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)
- [Pseudo-Hamiltonian neural networks for learning partial differential equations](https://mlsanity.com/paper/2304.14374/) (2023)
- [A Thermodynamics-informed Active Learning Approach to Perception and Reasoning about Fluids](https://mlsanity.com/paper/2203.05775/) (2022)
- [Neural oscillators for generalization of physics-informed machine learning](https://mlsanity.com/paper/2308.08989/) (2023)
- [Learning Neural Hamiltonian Dynamics: A Methodological Overview](https://mlsanity.com/paper/2203.00128/) (2022)
- [Thermodynamically Consistent Latent Dynamics Identification for Parametric Systems](https://mlsanity.com/paper/2506.08475/) (2025)
- [Physics-Informed Learning Using Hamiltonian Neural Networks with Output Error Noise Models](https://mlsanity.com/paper/2305.01338/) (2023)
- [Recovering implicit physics model under real-world constraints](https://mlsanity.com/paper/2412.02215/) (2024)
- [AdamNODEs: When Neural ODE Meets Adaptive Moment Estimation](https://mlsanity.com/paper/2207.06066/) (2022)
- [Thermodynamics-informed super-resolution of scarce temporal dynamics data](https://mlsanity.com/paper/2402.17506/) (2024)
- [Data-driven particle dynamics: Structure-preserving coarse-graining for emergent behavior in non-equilibrium systems](https://mlsanity.com/paper/2508.12569/) (2025)
- [Identifiable learning of dissipative dynamics](https://mlsanity.com/paper/2510.24160/) (2025)
- [Equilibrium Conserving Neural Operators for Super-Resolution Learning](https://mlsanity.com/paper/2504.13422/) (2025)
- [SPIEDiff: robust learning of long-time macroscopic dynamics from short-time particle simulations with quantified epistemic uncertainty](https://mlsanity.com/paper/2505.13501/) (2025)
- [Rapid training of Hamiltonian graph networks using random features](https://mlsanity.com/paper/2506.06558/) (2025)
- [Meta-learning Structure-Preserving Dynamics](https://mlsanity.com/paper/2508.11205/) (2025)
- [Metriplector: From Field Theory to Neural Architecture](https://mlsanity.com/paper/2603.29496/) (2026)

## Abstract (arXiv)

Forecasting of time-series data requires imposition of inductive biases to obtain predictive extrapolation, and recent works have imposed Hamiltonian/Lagrangian form to preserve structure for systems with reversible dynamics. In this work we present a novel parameterization of dissipative brackets from metriplectic dynamical systems appropriate for learning irreversible dynamics with unknown a priori model form. The process learns generalized Casimirs for energy and entropy guaranteed to be conserved and nondecreasing, respectively. Furthermore, for the case of added thermal noise, we guarantee exact preservation of a fluctuation-dissipation theorem, ensuring thermodynamic consistency. We provide benchmarks for dissipative systems demonstrating learned dynamics are more robust and generalize better than either "black-box" or penalty-based approaches.