Neural network from TENET exploiting time inversion

Let's first consider the causal open dynamical system model (1). If for the transition function we take an activation function applied to a linear state transition and add an output equation you may recognize a standard recurrent neural network (RNN), which is very well suited for the dynamical system modelling by construction and widely used to handle sequential data, e.g. It's quite a complicated model which requires taking into account both internal autonomous system state and external inputs. So let's simplify this complicated model by reformulating the problem to a more complex one-- consider the closed dynamical system. For that, we have to expand our internal system space by adding an external subsystem containing dynamics of the external variables. This tradeoff between the model complication and the problem complexity is not cheap and will cost us later when we teach the model to understand new variables' relations of the expanded internal space.