# Metriplectic Conditional Flow Matching for Dissipative Dynamics

arXiv: https://arxiv.org/abs/2509.19526 · HTML: https://mlsanity.com/paper/2509.19526/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 13 of its 20 references that are in the map.
- First posted on arXiv 23 Sep 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 13 of 13)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Variational Integrator Networks for Physically Structured Embeddings](https://mlsanity.com/paper/1910.09349/) (2019)

## Built on by (most cited first, 4 of 4)

- [Logic-Guided Vector Fields for Constrained Generative Modeling](https://mlsanity.com/paper/2602.02009/) (2026)
- [Metriplector: From Field Theory to Neural Architecture](https://mlsanity.com/paper/2603.29496/) (2026)
- [GENERIC-FNO: Embedding Energy Conservation and Entropy Production into Fourier Neural Operators](https://mlsanity.com/paper/2606.08343/) (2026)
- [Poisson-GENERIC Neural Operators: Exact Metriplectic Structure in Function Space via Casimir Entropies](https://mlsanity.com/paper/2610.05570/) (2026)

## Abstract (arXiv)

Metriplectic conditional flow matching (MCFM) learns dissipative dynamics without violating first principles. Neural surrogates often inject energy and destabilize long-horizon rollouts; MCFM instead builds the conservative-dissipative split into both the vector field and a structure preserving sampler. MCFM trains via conditional flow matching on short transitions, avoiding long rollout adjoints. In inference, a Strang-prox scheme alternates a symplectic update with a proximal metric step, ensuring discrete energy decay; an optional projection enforces strict decay when a trusted energy is available. We provide continuous and discrete time guarantees linking this parameterization and sampler to conservation, monotonic dissipation, and stable rollouts. On a controlled mechanical benchmark, MCFM yields phase portraits closer to ground truth and markedly fewer energy-increase and positive energy rate events than an equally expressive unconstrained neural flow, while matching terminal distributional fit.