  A deterministic temporal process can be determined by its trajectory, an
element in the product space of (a) initial condition $z_0 \in \mathcal{Z}$ and
(b) transition function $f: (\mathcal{Z}, \mathcal{T}) \to \mathcal{Z}$ often
influenced by the control of the underlying dynamical system. Existing methods
often model the transition function as a differential equation or as a
recurrent neural network. Despite their effectiveness in predicting future
measurements, few results have successfully established a method for sampling
and statistical inference of trajectories using neural networks, partially due
to constraints in the parameterization. In this work, we introduce a mechanism
to learn the distribution of trajectories by parameterizing the transition
function $f$ explicitly as an element in a function space. Our framework allows
efficient synthesis of novel trajectories, while also directly providing a
convenient tool for inference, i.e., uncertainty estimation, likelihood
evaluations and out of distribution detection for abnormal trajectories. These
capabilities can have implications for various downstream tasks, e.g.,
simulation and evaluation for reinforcement learning.
