  Encoding constraints into neural networks is attractive. This paper studies
how to introduce the popular positive linear satisfiability to neural networks.
We propose the first differentiable satisfiability layer based on an extension
of the classic Sinkhorn algorithm for jointly encoding multiple sets of
marginal distributions. We further theoretically characterize the convergence
property of the Sinkhorn algorithm for multiple marginals. In contrast to the
sequential decision e.g.\ reinforcement learning-based solvers, we showcase our
technique in solving constrained (specifically satisfiability) problems by
one-shot neural networks, including i) a neural routing solver learned without
supervision of optimal solutions; ii) a partial graph matching network handling
graphs with unmatchable outliers on both sides; iii) a predictive network for
financial portfolios with continuous constraints. To our knowledge, there
exists no one-shot neural solver for these scenarios when they are formulated
as satisfiability problems. Source code is available at
https://github.com/Thinklab-SJTU/LinSATNet
