Model order reduction has been extensively studied over the last two decades. Projection-based methods such as the Proper Orthogonal Decomposition and the Reduced Basis Method enjoy the important advantages of Galerkin methods in the derivation of the reduced problem, but are limited to linear data compression for which the reduced solution is sought as a linear combination of spatial modes. Nonlinear data compression must be used when the solution manifold is not embedded in a low-dimensional subspace. Early methods involve piecewise linear data compression, by constructing a dictionary of reduced-order models tailored to a partition of the solution manifold. In this work, we introduce the concept of optimal partition of the solution manifold in terms of normalized Kolmogorov widths, and prove that the optimal partitions can be found by means of a representative-based clustering algorithm using the sine dissimilarity measure on the solution manifold.

翻译：在过去二十年中,人们广泛研究了模型的减少,例如正正正正正正正正正正分解和减低底线法等基于预测的方法在逐渐减少问题的过程中享有Galerkin方法的重要优势,但仅限于线性数据压缩,为此,将缩小后的解决方案作为空间模式的线性组合来寻求。当解决方案的方位没有嵌入低维次空间时,必须使用非线性数据压缩。早期方法涉及小巧线性数据压缩,方法是建立一个适合解决方案方位分隔的减序模型词典。在这项工作中,我们引入了以正常的科尔莫戈罗夫宽度为标准对多个解决方案进行最佳分割的概念,并证明最佳分区可以通过在解决方案上采用正式差异计量法的有代表性的组合算法找到。