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.

翻译：我们提出了一种新颖的变分框架，用于对由马尔可夫近似分数布朗运动（fBM）驱动的（神经）随机微分方程（SDE）进行推断。SDE为建模具有内在噪声和随机性的真实世界连续时间动态系统提供了一种通用工具。将SDE与变分方法强大的推断能力相结合，可通过随机梯度下降学习代表性函数分布。然而，传统SDE通常假设潜在噪声遵循布朗运动（BM），这限制了其捕捉长期依赖性的能力。相比之下，分数布朗运动（fBM）将BM扩展至涵盖非马尔可夫动力学，但现有推断fBM参数的方法要么计算成本高昂，要么统计效率低下。本文基于fBM的马尔可夫近似，从成熟的随机分析领域推导出高效变分推断后验路径度量所需的下界证据。此外，我们提供了一种闭式表达式以确定最优近似系数。进一步地，我们提出使用神经网络学习变分后验中的漂移项、扩散项和控制项，从而实现神经SDE的变分训练。在该框架中，我们还优化了控制分数噪声性质的赫斯特指数。除了在合成数据上的验证，我们还贡献了一种用于变分潜在视频预测的新颖架构——据我们所知，该方法首次实现了视频感知领域的变分神经SDE应用。