  Understanding the dynamics of a system is important in many scientific and
engineering domains. This problem can be approached by learning state
transition rules from observations using machine learning techniques. Such
observed time-series data often consist of sequences of many continuous
variables with noise and ambiguity, but we often need rules of dynamics that
can be modeled with a few essential variables. In this work, we propose a
method for extracting a small number of essential hidden variables from
high-dimensional time-series data and for learning state transition rules
between these hidden variables. The proposed method is based on the Restricted
Boltzmann Machine (RBM), which treats observable data in the visible layer and
latent features in the hidden layer. However, real-world data, such as video
and audio, include both discrete and continuous variables, and these variables
have temporal relationships. Therefore, we propose Recurrent Temporal
GaussianBernoulli Restricted Boltzmann Machine (RTGB-RBM), which combines
Gaussian-Bernoulli Restricted Boltzmann Machine (GB-RBM) to handle continuous
visible variables, and Recurrent Temporal Restricted Boltzmann Machine (RT-RBM)
to capture time dependence between discrete hidden variables. We also propose a
rule-based method that extracts essential information as hidden variables and
represents state transition rules in interpretable form. We conduct experiments
on Bouncing Ball and Moving MNIST datasets to evaluate our proposed method.
Experimental results show that our method can learn the dynamics of those
physical systems as state transition rules between hidden variables and can
predict unobserved future states from observed state transitions.
