DFORM: Diffeomorphic vector field alignment for assessing dynamics
  across learned models
Ruiqi Chen Giacomo Vedovati Todd Braver ShiNung Ching
Dynamical system models such as Recurrent Neural Networks (RNNs) have become
increasingly popular as hypothesis-generating tools in scientific research.
Evaluating the dynamics in such networks is key to understanding their learned
generative mechanisms. However, comparison of learned dynamics across models is
challenging due to their inherent nonlinearity and because a priori there is no
enforced equivalence of their coordinate systems. Here, we propose the DFORM
(Diffeomorphic vector field alignment for comparing dynamics across learned
models) framework. DFORM learns a nonlinear coordinate transformation which
provides a continuous, maximally one-to-one mapping between the trajectories of
learned models, thus approximating a diffeomorphism between them. The mismatch
between DFORM-transformed vector fields defines the orbital similarity between
two models, thus providing a generalization of the concepts of smooth orbital
and topological equivalence. As an example, we apply DFORM to models trained on
a canonical neuroscience task, showing that learned dynamics may be
functionally similar, despite overt differences in attractor landscapes.