Surrogate models can reduce computational costs for multivariable functions with 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 to effectively solve this problem in the case of non-adaptive tensor recovery from a given training set (i.e., tensor completion problem). TT-ALS allows obtaining a low-parametric representation of the tensor, which is free from the curse of dimensionality and can be used for fast computation of the values at arbitrary tensor indices or efficient implementation of algebra operations with the black box (integration, etc.). However, to obtain high accuracy in the presence of restrictions on the size of the train data, a good choice of initial approximation is essential. In this work, we construct the ANOVA representation in the TT-format and use it as an initial approximation for the TT-ALS algorithm. The performed numerical computations for a number of multidimensional model problems, including the parametric partial differential equation, demonstrate a significant advantage of our approach for the commonly used random initial approximation. For all considered model problems we obtained an increase in accuracy by at least an order of magnitude with the same number of requests to the black box. The proposed approach is very general and can be applied in a wide class of real-world surrogate modeling and machine learning problems.
翻译:代理模型可以降低内部结构(黑盒)未知的多变量功能的计算成本。 在离散的配方中,替代模型相当于从一小部分元素中恢复一个多维阵列(tensor) 。 在高压列(TT) 格式中,交替最小正方(ALS)算法是一种广泛使用的方法,在对火车数据大小存在限制的情况下,为了有效地解决这一问题,必须从一个特定的培训组(即,单数完成问题)中选择良好的初始近似方法。TT-ALS 可以在TT-ALS 算法中构建一个低参数表示法,它不受维度诅咒,可用于快速计算任意高压指数中的值,或用于快速计算与黑盒(集成等)一道的代数,或高效执行代数操作的代数操作。然而,在对火车数据大小存在限制的情况下,选择初步近似的方法至关重要。 在这项工作中,我们在TT-ALS算法中构建ANOVA代表制,并使用它作为黑色算法的初始近度缩缩缩缩缩缩缩缩缩缩图。在计算模型中,在计算中,在计算中,在计算中,在计算中要用一个高度模型中,在计算中,在计算中,在计算中,在计算中,在采用一个用于测算法的多数平式模型中,在计算方法中,在计算,在计算,在计算时,在计算,在计算,在计算时,在计算,在计算,在计算,在计算,在计算,在计算时,在计算时,在计算,在计算,在计算时,在计算,在计算,在计算,在计算时,在计算,在计算中,在计算,在计算,在计算,在计算,在计算,在计算,在计算,在计算时,在计算时,在计算时,在计算,在计算时,在计算,在计算模型中,在计算时,在计算,在计算时,在计算时,在计算时,在计算,在计算,在计算时,在计算,在计算,在计算,在计算,在计算,在计算时,在计算时,在计算,在计算,在计算时,在计算,在计算,在计算,在计算,在计算时,在计算,在计算,在计算