NeurIPS 2025

SING: SDE Inference via Natural Gradients
Amber Hu∗
Stanford University
amberhu@stanford.edu
Henry D. Smith∗
Stanford University
smithhd@stanford.edu
Scott W. Linderman
Stanford University
swl1@stanford.edu
Abstract
Latent stochastic differential equation (SDE) models are important tools for the
unsupervised discovery of dynamical systems from data, with applications ranging
from engineering to neuroscience. In these complex domains, exact posterior
inference of the latent state path is typically intractable, motivating the use of
approximate methods such as variational inference (VI). However, existing VI
methods for inference in latent SDEs often suffer from slow convergence and
numerical instability. Here, we propose SDE Inference via Natural Gradients
(SING), a method that leverages natural gradient VI to efficiently exploit the un-
derlying geometry of the model and variational posterior. SING enables fast and
reliable inference in latent SDE models by approximating intractable integrals
and parallelizing computations in time. We provide theoretical guarantees that
SING approximately optimizes the intractable, continuous-time objective of inter-
est. Moreover, we demonstrate that better state inference enables more accurate
estimation of nonlinear drift functions using, for example, Gaussian process SDE
models. SING outperforms prior methods in state inference and drift estimation
on a variety of datasets, including a challenging application to modeling neural
dynamics in freely behaving animals. Altogether, our results illustrate the potential
of SING as a tool for accurate inference in complex dynamical systems, especially
those characterized by limited prior knowledge and non-conjugate structure.1
1
Introduction
Stochastic differential equations (SDEs) are powerful tools for modeling complex, time-varying
systems in scientific domains such as physics, engineering, and neuroscience [1, 2]. In many real-
world settings, we do not observe the state of the syste...