  We present a method for learning neural representations of flow maps from
time-varying vector field data. The flow map is pervasive within the area of
flow visualization, as it is foundational to numerous visualization techniques,
e.g. integral curve computation for pathlines or streaklines, as well as
computing separation/attraction structures within the flow field. Yet
bottlenecks in flow map computation, namely the numerical integration of vector
fields, can easily inhibit their use within interactive visualization settings.
In response, in our work we seek neural representations of flow maps that are
efficient to evaluate, while remaining scalable to optimize, both in
computation cost and data requirements. A key aspect of our approach is that we
can frame the process of representation learning not in optimizing for samples
of the flow map, but rather, a self-consistency criterion on flow map
derivatives that eliminates the need for flow map samples, and thus numerical
integration, altogether. Central to realizing this is a novel neural network
design for flow maps, coupled with an optimization scheme, wherein our
representation only requires the time-varying vector field for learning,
encoded as instantaneous velocity. We show the benefits of our method over
prior works in terms of accuracy and efficiency across a range of 2D and 3D
time-varying vector fields, while showing how our neural representation of flow
maps can benefit unsteady flow visualization techniques such as streaklines,
and the finite-time Lyapunov exponent.
