This paper proposes a probabilistic model of subspaces based on the probabilistic principal component analysis (PCA). Given a sample of vectors in the embedding space -- commonly known as a snapshot matrix -- this method uses quantities derived from the probabilistic PCA to construct distributions of the sample matrix, as well as the principal subspaces. It is applicable to projection-based reduced-order modeling methods, such as proper orthogonal decomposition and related model reduction methods. The stochastic subspace thus constructed can be used, for example, to characterize model-form uncertainty in computational mechanics. The proposed method has multiple desirable properties: (1) it is naturally justified by the probabilistic PCA and has analytic forms for the induced random matrix models; (2) it satisfies linear constraints, such as boundary conditions of all kinds, by default; (3) it has only one hyperparameter, which significantly simplifies training; and (4) its algorithm is very easy to implement. We demonstrate the performance of the proposed method via several numerical examples in computational mechanics and structural dynamics.

翻译：本文提出了一种基于概率主成分分析（PCA）的子空间概率模型。给定嵌入空间中的向量样本（通常称为快照矩阵），该方法利用概率PCA导出的量来构建样本矩阵及主子空间的分布。它适用于基于投影的降阶建模方法，例如本征正交分解及相关模型降阶方法。由此构建的随机子空间可用于表征计算力学中的模型形式不确定性。所提方法具有多项理想特性：（1）其概率基础自然源自概率PCA，且诱导的随机矩阵模型具有解析形式；（2）默认满足各类线性约束（如所有类型的边界条件）；（3）仅包含一个超参数，极大简化了训练过程；（4）算法实现极为简便。我们通过计算力学和结构动力学中的若干数值算例验证了所提方法的性能。