Variational Inference for (Neural) SDEs Driven by Fractional Noise

Stochastic differential equations (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 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. Building upon the Markov approximation of fBM, we derive the evidence lower bound essential for efficient variational inference of posterior path measures, leading to the variational training of neural-SDEs driven by fBM. Beyond validation on synthetic data, we contribute a novel architecture for variational latent video prediction.