Robust iterative methods for solving large sparse systems of linear algebraic equations often suffer from the problem of optimizing the corresponding tuning parameters. To improve the performance for the problem of interest, the specific parameter tuning is required, which in practice can be a timeconsuming and tedious task. The present paper deals with the problem of automating the optimization of the numerical method parameters to improve the performance of the mathematical physics simulations and simplify the modeling process.The paper proposes the hybrid evolution strategy applied to tune the parameters of the Krylov subspace and algebraic multigrid iterative methods when solving a sequence of linear systems with a constant matrix and varying right-hand side. The algorithm combines the traditional evolution strategy with the pre-trained neural network, which filters the individuals in the new generation. The proposed coupling of two optimization approaches allows to integrate the adaptivity properties of the evolution strategy with a priori knowledge realized by the neural network. Furthermore, the use of the neural network as a preliminary filter allows for significant weakening of the prediction accuracy requirements and reusing the pre-trained network with a wide range of linear systems.The detailed algorithm efficiency evaluation is performed for a set of model linear systems, including the ones from the SuiteSparse Matrix Collection and the systems from the turbulent flow simulations. The obtained results show that the pre-trained neural network can be effectively reused to optimize parameters for various linear systems, and a significant speedup in the calculations can be achieved at the cost of about 100 trial solves. The algorithm decreases the calculation time by more than 6 times for the black box matrices from the SuiteSparse Matrix Collection and by a factor of 1.5-1.8 for the turbulent flow simulations considered in the paper.

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