# GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems

arXiv: https://arxiv.org/abs/2109.00092 · HTML: https://mlsanity.com/paper/2109.00092/

## Key facts

- Cited by 49 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 46 references that are in the map.
- First posted on arXiv 31 Aug 2021.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Generative Modeling using the Sliced Wasserstein Distance](https://mlsanity.com/paper/1803.11188/) (2018)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Reversible Architectures for Arbitrarily Deep Residual Neural Networks](https://mlsanity.com/paper/1709.03698/) (2017)
- [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)
- [Deep learning of thermodynamics-aware reduced-order models from data](https://mlsanity.com/paper/2007.03758/) (2020)
- [Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks](https://mlsanity.com/paper/2012.03133/) (2020)
- [OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle](https://mlsanity.com/paper/2009.02327/) (2020)
- [Maximum Likelihood Estimation for Stochastic Differential Equations Using Sequential Kriging-Based Optimization](https://mlsanity.com/paper/1408.2441/) (2014)

## Built on by (most cited first, 30 of 49)

- [Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systems](https://mlsanity.com/paper/2205.08304/) (2022)
- [Thermodynamics-informed graph neural networks](https://mlsanity.com/paper/2203.01874/) (2022)
- [Correcting model misspecification in physics-informed neural networks (PINNs)](https://mlsanity.com/paper/2310.10776/) (2023)
- [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)
- [Efficiently Parameterized Neural Metriplectic Systems](https://mlsanity.com/paper/2405.16305/) (2024)
- [Pseudo-Hamiltonian Neural Networks with State-Dependent External Forces](https://mlsanity.com/paper/2206.02660/) (2022)
- [Exact conservation laws for neural network integrators of dynamical systems](https://mlsanity.com/paper/2209.11661/) (2022)
- [Stable Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2502.02480/) (2025)
- [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)
- [MeshGraphNet-Transformer: Scalable Mesh-based Learned Simulation for Solid Mechanics](https://mlsanity.com/paper/2601.23177/) (2026)
- [SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems](https://mlsanity.com/paper/2201.05475/) (2022)
- [A Thermodynamics-informed Active Learning Approach to Perception and Reasoning about Fluids](https://mlsanity.com/paper/2203.05775/) (2022)
- [Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries](https://mlsanity.com/paper/2308.15349/) (2023)
- [DynGMA: a robust approach for learning stochastic differential equations from data](https://mlsanity.com/paper/2402.14475/) (2024)
- [Thermodynamically Consistent Latent Dynamics Identification for Parametric Systems](https://mlsanity.com/paper/2506.08475/) (2025)
- [Implementation and (Inverse Modified) Error Analysis for implicitly-templated ODE-nets](https://mlsanity.com/paper/2303.17824/) (2023)
- [Physics-Aware Neural Implicit Solvers for multiscale, parametric PDEs with applications in heterogeneous media](https://mlsanity.com/paper/2405.19019/) (2024)
- [SympGNNs: Symplectic Graph Neural Networks for identifiying high-dimensional Hamiltonian systems and node classification](https://mlsanity.com/paper/2408.16698/) (2024)
- [Thermodynamics-informed super-resolution of scarce temporal dynamics data](https://mlsanity.com/paper/2402.17506/) (2024)
- [Learning Macroscopic Dynamics from Partial Microscopic Observations](https://mlsanity.com/paper/2410.23938/) (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)

## Abstract (arXiv)

We propose the GENERIC formalism informed neural networks (GFINNs) that obey the symmetric degeneracy conditions of the GENERIC formalism. GFINNs comprise two modules, each of which contains two components. We model each component using a neural network whose architecture is designed to satisfy the required conditions. The component-wise architecture design provides flexible ways of leveraging available physics information into neural networks. We prove theoretically that GFINNs are sufficiently expressive to learn the underlying equations, hence establishing the universal approximation theorem. We demonstrate the performance of GFINNs in three simulation problems: gas containers exchanging heat and volume, thermoelastic double pendulum and the Langevin dynamics. In all the examples, GFINNs outperform existing methods, hence demonstrating good accuracy in predictions for both deterministic and stochastic systems.