  We consider the problem of simulating diffusion bridges, which are diffusion
processes that are conditioned to initialize and terminate at two given states.
The simulation of diffusion bridges has applications in diverse scientific
fields and plays a crucial role in the statistical inference of
discretely-observed diffusions. This is known to be a challenging problem that
has received much attention in the last two decades. This article contributes
to this rich body of literature by presenting a new avenue to obtain diffusion
bridge approximations. Our approach is based on a backward time representation
of a diffusion bridge, which may be simulated if one can time-reverse the
unconditioned diffusion. We introduce a variational formulation to learn this
time-reversal with function approximation and rely on a score matching method
to circumvent intractability. Another iteration of our proposed methodology
approximates the Doob's $h$-transform defining the forward time representation
of a diffusion bridge. We discuss algorithmic considerations and extensions,
and present numerical results on an Ornstein--Uhlenbeck process, a model from
financial econometrics for interest rates, and a model from genetics for cell
differentiation and development to illustrate the effectiveness of our
approach.
