# Gaussian Flow Dynamics: Simulation-Free Neural SDE Learning Beyond One-Time Marginals

arXiv: https://arxiv.org/abs/2610.04390 · HTML: https://mlsanity.com/paper/2610.04390/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 47 references that are in the map.
- First posted on arXiv 3 Oct 2026.

## Builds on (most cited first, 30 of 32)

- [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)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [Diffusion Models With Learned Adaptive Noise](https://mlsanity.com/paper/2312.13236/) (2023)
- [Neural Flow Diffusion Models: Learnable Forward Process for Improved Diffusion Modelling](https://mlsanity.com/paper/2404.12940/) (2024)
- [Improving Variational Auto-Encoders using Householder Flow](https://mlsanity.com/paper/1611.09630/) (2016)
- [Learning interpretable continuous-time models of latent stochastic dynamical systems](https://mlsanity.com/paper/1902.04420/) (2019)
- [Where to Diffuse, How to Diffuse, and How to Get Back: Automated Learning for Multivariate Diffusions](https://mlsanity.com/paper/2302.07261/) (2023)
- [Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Sampling](https://mlsanity.com/paper/2410.07974/) (2024)
- [Black-box Variational Inference for Stochastic Differential Equations](https://mlsanity.com/paper/1802.03335/) (2018)
- [On gauge freedom, conservativity and intrinsic dimensionality estimation in diffusion models](https://mlsanity.com/paper/2402.03845/) (2024)
- [SDE Matching: Scalable and Simulation-Free Training of Latent Stochastic Differential Equations](https://mlsanity.com/paper/2502.02472/) (2025)
- [Neural Diffusion Models](https://mlsanity.com/paper/2310.08337/) (2023)
- [Curly Flow Matching for Learning Non-gradient Field Dynamics](https://mlsanity.com/paper/2510.26645/) (2025)
- [Amortized Reparametrization: Efficient and Scalable Variational Inference for Latent SDEs](https://mlsanity.com/paper/2312.10550/) (2023)
- [DiffEnc: Variational Diffusion with a Learned Encoder](https://mlsanity.com/paper/2310.19789/) (2023)
- [Deep Temporal Sigmoid Belief Networks for Sequence Modeling](https://mlsanity.com/paper/1509.07087/) (2015)
- [Gaussian processes at the Helm(holtz): A more fluid model for ocean currents](https://mlsanity.com/paper/2302.10364/) (2023)
- [Variational Gaussian Process Diffusion Processes](https://mlsanity.com/paper/2306.02066/) (2023)
- [A variational approach to path estimation and parameter inference of hidden diffusion processes](https://mlsanity.com/paper/1508.00506/) (2015)
- [Variational Inference for Continuous-Time Switching Dynamical Systems](https://mlsanity.com/paper/2109.14492/) (2021)
- [Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series](https://mlsanity.com/paper/2410.05602/) (2024)
- [Moment-Based Variational Inference for Stochastic Differential Equations](https://mlsanity.com/paper/2103.00988/) (2021)
- [SING: SDE Inference via Natural Gradients](https://mlsanity.com/paper/2506.17796/) (2025)
- [Neural Stochastic Flows: Solver-Free Modelling and Inference for SDE Solutions](https://mlsanity.com/paper/2510.25769/) (2025)

## Abstract (arXiv)

Simulation-free training of latent Stochastic Differential Equations (SDEs) relies on a variational posterior process whose one-time marginals are tractable, typically Gaussian. Such marginals, however, do not determine the underlying dynamics: many processes share the same marginals while differing in their temporal structure, and existing parameterizations fix this structure implicitly, which restricts the posterior family and biases the learned model. We introduce Gaussian flow dynamics, which construct stochastic processes directly from smoothly evolving Gaussian marginals while making the marginal-preserving, or gauge, degrees of freedom explicit and parameterizable. The construction admits state-dependent diffusion coefficients and recovers every linear SDE with additive noise and a non-degenerate Gaussian initial distribution. Building on it, we propose Gauge Matching, a simulation-free method for latent SDE learning that combines Gaussian flow dynamics with the SDE Matching objective. Gauge Matching costs at most quadratically in the latent dimension per step, like SDE Matching, but learns the temporal structure of the posterior beyond its one-time marginals. It comes within a nat of Helmholtz-SDE, which computes the gauge from the prior Jacobian at cubic cost, on the linear benchmark where the exact posterior is known, matches it on nonlinear systems, and applies where Helmholtz-SDE does not, to state-dependent noise.