# Variational Inference for SDEs Driven by Fractional Noise

arXiv: https://arxiv.org/abs/2310.12975 · HTML: https://mlsanity.com/paper/2310.12975/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 48 references that are in the map.
- First posted on arXiv 19 Oct 2023.

## Builds on (most cited first, 22 of 22)

- [VideoFusion: Decomposed Diffusion Models for High-Quality Video Generation](https://mlsanity.com/paper/2303.08320/) (2023)
- [HuMoR: 3D Human Motion Model for Robust Pose Estimation](https://mlsanity.com/paper/2105.04668/) (2021)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Stochastic Video Generation with a Learned Prior](https://mlsanity.com/paper/1802.07687/) (2018)
- [Diffusion Probabilistic Modeling for Video Generation](https://mlsanity.com/paper/2203.09481/) (2022)
- [Stochastic Variational Video Prediction](https://mlsanity.com/paper/1710.11252/) (2017)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [Stochastic Latent Residual Video Prediction](https://mlsanity.com/paper/2002.09219/) (2020)
- [Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise](https://mlsanity.com/paper/1906.02355/) (2019)
- [SDE-Net: Equipping Deep Neural Networks with Uncertainty Estimates](https://mlsanity.com/paper/2008.10546/) (2020)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [Weakly Supervised Learning of Rigid 3D Scene Flow](https://mlsanity.com/paper/2102.08945/) (2021)
- [CaSPR: Learning Canonical Spatiotemporal Point Cloud Representations](https://mlsanity.com/paper/2008.02792/) (2020)
- [Vid-ODE: Continuous-Time Video Generation with Neural Ordinary Differential Equation](https://mlsanity.com/paper/2010.08188/) (2020)
- [Black-box Variational Inference for Stochastic Differential Equations](https://mlsanity.com/paper/1802.03335/) (2018)
- [Adaptive importance sampling for control and inference](https://mlsanity.com/paper/1505.01874/) (2015)
- [Neural network stochastic differential equation models with applications to financial data forecasting](https://mlsanity.com/paper/2111.13164/) (2021)
- [Learning stochastic differential equations using RNN with log signature features](https://mlsanity.com/paper/1908.08286/) (2019)
- [Latent Neural Differential Equations for Video Generation](https://mlsanity.com/paper/2011.03864/) (2020)
- [Fractional SDE-Net: Generation of Time Series Data with Long-term Memory](https://mlsanity.com/paper/2201.05974/) (2022)
- [Statistical Inference for Stochastic Differential Equations with Memory](https://mlsanity.com/paper/1307.1164/) (2013)
- [VidStyleODE: Disentangled Video Editing via StyleGAN and NeuralODEs](https://mlsanity.com/paper/2304.06020/) (2023)

## Built on by (most cited first, 9 of 9)

- [SDE Matching: Scalable and Simulation-Free Training of Latent Stochastic Differential Equations](https://mlsanity.com/paper/2502.02472/) (2025)
- [From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster training](https://mlsanity.com/paper/2501.06148/) (2025)
- [Generative Fractional Diffusion Models](https://mlsanity.com/paper/2310.17638/) (2023)
- [Neural Stochastic Flows: Solver-Free Modelling and Inference for SDE Solutions](https://mlsanity.com/paper/2510.25769/) (2025)
- [Efficient Training of Neural SDEs Using Stochastic Optimal Control](https://mlsanity.com/paper/2505.17150/) (2025)
- [Fractional Diffusion Bridge Models](https://mlsanity.com/paper/2511.01795/) (2025)
- [ABC: Any-Subset Autoregression via Non-Markovian Diffusion Bridges in Continuous Time and Space](https://mlsanity.com/paper/2604.27443/) (2026)
- [Variational Inference for L'evy Process-Driven SDEs via Neural Tilting](https://mlsanity.com/paper/2605.10934/) (2026)
- [Continuous-Time Trajectory Generation from Discrete Observations with Stochasticity](https://mlsanity.com/paper/2609.32026/) (2026)

## Abstract (arXiv)

We present a novel variational framework for performing inference in (neural) stochastic differential equations (SDEs) driven by Markov-approximate fractional Brownian motion (fBM). SDEs offer a versatile tool for modeling real-world continuous-time dynamic systems with inherent noise and randomness. Combining SDEs with the powerful inference capabilities of variational methods, enables the learning of representative function distributions through stochastic gradient descent. However, conventional SDEs typically assume the underlying noise to follow a Brownian motion (BM), which hinders their ability to capture long-term dependencies. In contrast, fractional Brownian motion (fBM) extends BM to encompass non-Markovian dynamics, but existing methods for inferring fBM parameters are either computationally demanding or statistically inefficient. In this paper, building upon the Markov approximation of fBM, we derive the evidence lower bound essential for efficient variational inference of posterior path measures, drawing from the well-established field of stochastic analysis. Additionally, we provide a closed-form expression to determine optimal approximation coefficients. Furthermore, we propose the use of neural networks to learn the drift, diffusion and control terms within our variational posterior, leading to the variational training of neural-SDEs. In this framework, we also optimize the Hurst index, governing the nature of our fractional noise. Beyond validation on synthetic data, we contribute a novel architecture for variational latent video prediction,-an approach that, to the best of our knowledge, enables the first variational neural-SDE application to video perception.