Therefore, they seemtobegoodcandidatestobuild SSPsolutions, since Property 1 statesthattheidealsolverU is permutation-equivariant (thiswillbeconfirmedby Corollary 1).

This paper introduces Deep Statistical Solvers (DSS), a new class of trainable solvers for optimization problems, arising e.g., from system simulations. The key idea is to learn a solver that generalizes to a given distribution of problem instances. This is achieved by directly using as loss the objective function of the problem, as opposed to most previous Machine Learning based approaches, which mimic the solutions attained by an existing solver. Though both types of approaches outperform classical solvers with respect to speed for a given accuracy, a distinctive advantage of DSS is that they can be trained without a training set of sample solutions.

This paper introduces Deep Statistical Solvers (DSS), a new class of trainable solvers for optimization problems, arising e.g., from system simulations. The key idea is to learn a solver that generalizes to a given distribution of problem instances. This is achieved by directly using as loss the objective function of the problem, as opposed to most previous Machine Learning based approaches, which mimic the solutions attained by an existing solver.

The paper proposes new theoretical results regarding universal approximation property of graph convolutional neural networks and uses and trains them for (approximately) solving optimization problems defined on graphs, in particular arising from a discretization of PDEs. The solver is trained directly from the model energy. The paper was recognized by reviewers as having an interesting contribution and meeting the quality standards. The authors are invited to submit the final version including the rebuttal points, addressing all minor revision issues and the literature connections mentioned. Showing the applicability boundaries by studying failure cases is also highly appreciated.

This paper introduces Deep Statistical Solvers (DSS), a new class of trainable solvers for optimization problems, arising e.g., from system simulations. The key idea is to learn a solver that generalizes to a given distribution of problem instances. This is achieved by directly using as loss the objective function of the problem, as opposed to most previous Machine Learning based approaches, which mimic the solutions attained by an existing solver. Though both types of approaches outperform classical solvers with respect to speed for a given accuracy, a distinctive advantage of DSS is that they can be trained without a training set of sample solutions. Under sufficient conditions, we prove that the corresponding set of functions contains approximations to any arbitrary precision of the actual solution of the optimization problem.