# Black-box Variational Inference for Stochastic Differential Equations

arXiv: https://arxiv.org/abs/1802.03335 · HTML: https://mlsanity.com/paper/1802.03335/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2018.

## Built on by (most cited first, 25 of 25)

- [Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equations](https://mlsanity.com/paper/2102.06559/) (2021)
- [SDE Matching: Scalable and Simulation-Free Training of Latent Stochastic Differential Equations](https://mlsanity.com/paper/2502.02472/) (2025)
- [Batch and match: black-box variational inference with a score-based divergence](https://mlsanity.com/paper/2402.14758/) (2024)
- [Variational Gaussian Process Diffusion Processes](https://mlsanity.com/paper/2306.02066/) (2023)
- [Variational Inference for SDEs Driven by Fractional Noise](https://mlsanity.com/paper/2310.12975/) (2023)
- [An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable L'evy noise](https://mlsanity.com/paper/2201.13114/) (2022)
- [Longitudinal Flow Matching for Trajectory Modeling](https://mlsanity.com/paper/2510.03569/) (2025)
- [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)
- [Transformers Can Solve Non-Linear and Non-Markovian Filtering Problems in Continuous Time For Conditionally Gaussian Signals](https://mlsanity.com/paper/2310.19603/) (2023)
- [Learning to Suggest Breaks: Sustainable Optimization of Long-Term User Engagement](https://mlsanity.com/paper/2211.13585/) (2022)
- [TaylorSwiftNet: Taylor Driven Temporal Modeling for Swift Future Frame Prediction](https://mlsanity.com/paper/2110.14392/) (2021)
- [Wildfire Forecasting with Satellite Images and Deep Generative Model](https://mlsanity.com/paper/2208.09411/) (2022)
- [An information field theory approach to Bayesian state and parameter estimation in dynamical systems](https://mlsanity.com/paper/2306.02150/) (2023)
- [Learning Stochastic Hamiltonian Systems via Stochastic Generating Function Neural Network](https://mlsanity.com/paper/2507.14467/) (2025)
- [Large-scale Score-based Variational Posterior Inference for Bayesian Deep Neural Networks](https://mlsanity.com/paper/2602.05873/) (2026)
- [Neural Diffusion Intensity Models for Point Process Data](https://mlsanity.com/paper/2602.24083/) (2026)
- [Neural Galerkin Normalizing Flows for Bayesian Inference of Diffusions with Inaccessible Boundaries](https://mlsanity.com/paper/2606.04324/) (2026)
- [First-Order Trajectory Matching: Fast Ensemble Predictions of Chaotic, Turbulent, Stochastic Systems](https://mlsanity.com/paper/2606.11138/) (2026)
- [E2V-SDE: From Asynchronous Events to Fast and Continuous Video Reconstruction via Neural Stochastic Differential Equations](https://mlsanity.com/paper/2206.07578/) (2022)
- [FMint-SDE: A Multimodal Foundation Model for Accelerating Numerical Simulation of SDEs via Error Correction](https://mlsanity.com/paper/2510.27173/) (2025)
- [Multivariate Variational Autoencoder](https://mlsanity.com/paper/2511.07472/) (2025)
- [Variational Inference for L'evy Process-Driven SDEs via Neural Tilting](https://mlsanity.com/paper/2605.10934/) (2026)
- [PR-Smoother: Simulator-Preserving Non-Gaussian Smoothing for Data Assimilation](https://mlsanity.com/paper/2609.26890/) (2026)
- [Gaussian Flow Dynamics: Simulation-Free Neural SDE Learning Beyond One-Time Marginals](https://mlsanity.com/paper/2610.04390/) (2026)

## Abstract (arXiv)

Parameter inference for stochastic differential equations is challenging due to the presence of a latent diffusion process. Working with an Euler-Maruyama discretisation for the diffusion, we use variational inference to jointly learn the parameters and the diffusion paths. We use a standard mean-field variational approximation of the parameter posterior, and introduce a recurrent neural network to approximate the posterior for the diffusion paths conditional on the parameters. This neural network learns how to provide Gaussian state transitions which bridge between observations in a very similar way to the conditioned diffusion process. The resulting black-box inference method can be applied to any SDE system with light tuning requirements. We illustrate the method on a Lotka-Volterra system and an epidemic model, producing accurate parameter estimates in a few hours.