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Surrogate models can reduce computational costs for multivariable functionswith an unknown internal structure (black boxes). In a discrete formulation,surrogate modeling is equivalent to restoring a multidimensional array (tensor)from a small part of its elements. The alternating least squares (ALS)algorithm in the tensor train (TT) format is a widely used approach toeffectively solve this problem in the case of non-adaptive tensor recovery froma given training set (i.e., tensor completion problem). TT-ALS allows obtaininga low-parametric representation of the tensor, which is free from the curse ofdimensionality and can be used for fast computation of the values at arbitrarytensor indices or efficient implementation of algebra operations with the blackbox (integration, etc.). However, to obtain high accuracy in the presence ofrestrictions on the size of the train data, a good choice of initialapproximation is essential. In this work, we construct the ANOVA representationin the TT-format and use it as an initial approximation for the TT-ALSalgorithm. The performed numerical computations for a number ofmultidimensional model problems, including the parametric partial differentialequation, demonstrate a significant advantage of our approach for the commonlyused random initial approximation. For all considered model problems weobtained an increase in accuracy by at least an order of magnitude with thesame number of requests to the black box. The proposed approach is very generaland can be applied in a wide class of real-world surrogate modeling and machinelearning problems.

Black box approximation in the tensor train format initialized by ANOVA decomposition