  We develop a Bayesian inference method for discretely-observed stochastic
differential equations (SDEs). Inference is challenging for most SDEs, due to
the analytical intractability of the likelihood function. Nevertheless, forward
simulation via numerical methods is straightforward, motivating the use of
approximate Bayesian computation (ABC). We propose a conditional simulation
scheme for SDEs that is based on lookahead strategies for sequential Monte
Carlo (SMC) and particle smoothing using backward simulation. This leads to the
simulation of trajectories that are consistent with the observed trajectory,
thereby increasing the ABC acceptance rate. We additionally employ an invariant
neural network, previously developed for Markov processes, to learn the summary
statistics function required in ABC. The neural network is incrementally
retrained by exploiting an ABC-SMC sampler, which provides new training data at
each round. Since the SDEs simulation scheme differs from standard forward
simulation, we propose a suitable importance sampling correction, which has the
added advantage of guiding the parameters towards regions of high posterior
density, especially in the first ABC-SMC round. Our approach achieves accurate
inference and is about three times faster than standard (forward-only) ABC-SMC.
We illustrate our method in five simulation studies, including three examples
from the Chan-Karaolyi-Longstaff-Sanders SDE family, a stochastic bi-stable
model (Schl{\"o}gl) that is notoriously challenging for ABC methods, and a two
dimensional biochemical reaction network.
