  We study strategy synthesis for partially observable Markov decision
processes (POMDPs). The particular problem is to determine strategies that
provably adhere to (probabilistic) temporal logic constraints. This problem is
computationally intractable and theoretically hard. We propose a novel method
that combines techniques from machine learning and formal verification. First,
we train a recurrent neural network (RNN) to encode POMDP strategies. The RNN
accounts for memory-based decisions without the need to expand the full belief
space of a POMDP. Secondly, we restrict the RNN-based strategy to represent a
finite-memory strategy and implement it on a specific POMDP. For the resulting
finite Markov chain, efficient formal verification techniques provide provable
guarantees against temporal logic specifications. If the specification is not
satisfied, counterexamples supply diagnostic information. We use this
information to improve the strategy by iteratively training the RNN. Numerical
experiments show that the proposed method elevates the state of the art in
POMDP solving by up to three orders of magnitude in terms of solving times and
model sizes.
