  Several problems in stochastic analysis are defined through their geometry,
and preserving that geometric structure is essential to generating meaningful
predictions. Nevertheless, how to design principled deep learning (DL) models
capable of encoding these geometric structures remains largely unknown. We
address this open problem by introducing a universal causal geometric DL
framework in which the user specifies a suitable pair of metric spaces
$\mathscr{X}$ and $\mathscr{Y}$ and our framework returns a DL model capable of
causally approximating any ``regular'' map sending time series in
$\mathscr{X}^{\mathbb{Z}}$ to time series in $\mathscr{Y}^{\mathbb{Z}}$ while
respecting their forward flow of information throughout time. Suitable
geometries on $\mathscr{Y}$ include various (adapted) Wasserstein spaces
arising in optimal stopping problems, a variety of statistical manifolds
describing the conditional distribution of continuous-time finite state Markov
chains, and all Fr\'{e}chet spaces admitting a Schauder basis, e.g. as in
classical finance. Suitable spaces $\mathscr{X}$ are compact subsets of any
Euclidean space. Our results all quantitatively express the number of
parameters needed for our DL model to achieve a given approximation error as a
function of the target map's regularity and the geometric structure both of
$\mathscr{X}$ and of $\mathscr{Y}$. Even when omitting any temporal structure,
our universal approximation theorems are the first guarantees that H\"{o}lder
functions, defined between such $\mathscr{X}$ and $\mathscr{Y}$ can be
approximated by DL models.
