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We present a novel variational framework for performing inference in (neural)stochastic differential equations (SDEs) driven by Markov-approximatefractional Brownian motion (fBM). SDEs offer a versatile tool for modelingreal-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 throughstochastic gradient descent. However, conventional SDEs typically assume theunderlying noise to follow a Brownian motion (BM), which hinders their abilityto capture long-term dependencies. In contrast, fractional Brownian motion(fBM) extends BM to encompass non-Markovian dynamics, but existing methods forinferring fBM parameters are either computationally demanding or statisticallyinefficient. In this paper, building upon the Markov approximation of fBM, wederive the evidence lower bound essential for efficient variational inferenceof posterior path measures, drawing from the well-established field ofstochastic analysis. Additionally, we provide a closed-form expression todetermine optimal approximation coefficients. Furthermore, we propose the useof neural networks to learn the drift, diffusion and control terms within ourvariational posterior, leading to the variational training of neural-SDEs. Inthis framework, we also optimize the Hurst index, governing the nature of ourfractional noise. Beyond validation on synthetic data, we contribute a novelarchitecture for variational latent video prediction,-an approach that, to thebest of our knowledge, enables the first variational neural-SDE application tovideo perception.

Variational Inference for SDEs Driven by Fractional Noise