  We introduce Deep Reasoning Networks (DRNets), an end-to-end framework that
combines deep learning with reasoning for solving complex tasks, typically in
an unsupervised or weakly-supervised setting. DRNets exploit problem structure
and prior knowledge by tightly combining logic and constraint reasoning with
stochastic-gradient-based neural network optimization. We illustrate the power
of DRNets on de-mixing overlapping hand-written Sudokus (Multi-MNIST-Sudoku)
and on a substantially more complex task in scientific discovery that concerns
inferring crystal structures of materials from X-ray diffraction data under
thermodynamic rules (Crystal-Structure-Phase-Mapping). At a high level, DRNets
encode a structured latent space of the input data, which is constrained to
adhere to prior knowledge by a reasoning module. The structured latent encoding
is used by a generative decoder to generate the targeted output. Finally, an
overall objective combines responses from the generative decoder (thinking
fast) and the reasoning module (thinking slow), which is optimized using
constraint-aware stochastic gradient descent. We show how to encode different
tasks as DRNets and demonstrate DRNets' effectiveness with detailed
experiments: DRNets significantly outperform the state of the art and experts'
capabilities on Crystal-Structure-Phase-Mapping, recovering more precise and
physically meaningful crystal structures. On Multi-MNIST-Sudoku, DRNets
perfectly recovered the mixed Sudokus' digits, with 100% digit accuracy,
outperforming the supervised state-of-the-art MNIST de-mixing models. Finally,
as a proof of concept, we also show how DRNets can solve standard combinatorial
problems -- 9-by-9 Sudoku puzzles and Boolean satisfiability problems (SAT),
outperforming other specialized deep learning models. DRNets are general and
can be adapted and expanded to tackle other tasks.
