# Closing the Approximation Gap in Simulation-free Latent SDEs

arXiv: https://arxiv.org/abs/2606.16138 · HTML: https://mlsanity.com/paper/2606.16138/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 75 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 5 of 5)

- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Project and Generate: Divergence-Free Neural Operators for Incompressible Flows](https://mlsanity.com/paper/2603.24500/) (2026)
- [Universal Differential Equations as a Common Modeling Language for Neuroscience](https://mlsanity.com/paper/2403.14510/) (2024)
- [Probabilistic Inference and Learning with Stein's Method](https://mlsanity.com/paper/2603.07467/) (2026)
- [HDNet: Physics-Inspired Neural Network for Flow Estimation based on Helmholtz Decomposition](https://mlsanity.com/paper/2406.08570/) (2024)

## Built on by (most cited first, 2 of 2)

- [Gaussian Flow Dynamics: Simulation-Free Neural SDE Learning Beyond One-Time Marginals](https://mlsanity.com/paper/2610.04390/) (2026)
- [Scan-Parallel Variational Inference for event-driven Latent Jump Models](https://mlsanity.com/paper/2610.12986/) (2026)

## Abstract (arXiv)

Recovering dynamical systems from noisy observations is a recurring challenge across scientific domains, including neuroscience and physics. Latent stochastic differential equations (SDEs) address this by modeling the system as an unobserved state that evolves according to a learnable SDE and generates the observations. Variational inference (VI) provides a tractable objective for fitting latent SDEs. Traditional VI algorithms evaluate this objective by numerical simulation over a time discretization, trading fidelity for computational cost. A recent class of algorithms, simulation-free VI, sidesteps this tradeoff by parameterizing the posterior through its instantaneous marginals rather than its drift. In this work, we show that the efficiency of existing simulation-free VI algorithms comes at a price: their parameterizations restrict the approximate posterior to a subset of the SDEs available to simulation-based methods, degrading posterior inference and parameter learning. We propose Helmholtz-SDE, a simulation-free VI algorithm that closes this gap by optimizing over path laws compatible with a prescribed collection of marginals. Helmholtz-SDE recovers dynamics more faithfully than prior simulation-free methods, with the largest gains under high posterior uncertainty. It further matches the performance of simulation-based VI at a fraction of the runtime. Code is available at https://github.com/lindermanlab/helmholtz-sde .