  In this paper, we demonstrate how deterministic and stochastic dynamics on
manifolds, as well as differential geometric constructions can be implemented
concisely and efficiently using modern computational frameworks that mix
symbolic expressions with efficient numerical computations. In particular, we
use the symbolic expression and automatic differentiation features of the
python library Theano, originally developed for high-performance computations
in deep learning. We show how various aspects of differential geometry and Lie
group theory, connections, metrics, curvature, left/right invariance, geodesics
and parallel transport can be formulated with Theano using the automatic
computation of derivatives of any order. We will also show how symbolic
stochastic integrators and concepts from non-linear statistics can be
formulated and optimized with only a few lines of code. We will then give
explicit examples on low-dimensional classical manifolds for visualization and
demonstrate how this approach allows both a concise implementation and
efficient scaling to high dimensional problems.
