Human intelligence has shown remarkably lower latency and higher precision than most AI systems when processing non-stationary streaming data in real-time. Numerous neuroscience studies suggest that such abilities may be driven by internal predictive modeling. In this paper, we explore the possibility of introducing such a mechanism in unsupervised domain adaptation (UDA) for handling non-stationary streaming data for real-time streaming applications. We propose to formulate internal predictive modeling as a continuous-time Bayesian filtering problem within a stochastic dynamical system context. Such a dynamical system describes the dynamics of model parameters of a UDA model evolving with non-stationary streaming data. Building on such a dynamical system, we then develop extrapolative continuous-time Bayesian neural networks (ECBNN), which generalize existing Bayesian neural networks to represent temporal dynamics and allow us to extrapolate the distribution of model parameters before observing the incoming data, therefore effectively reducing the latency. Remarkably, our empirical results show that ECBNN is capable of continuously generating better distributions of model parameters along the time axis given historical data only, thereby achieving (1) training-free test-time adaptation with low latency, (2) gradually improved alignment between the source and target features and (3) gradually improved model performance over time during the real-time testing stage.