Modern day engineering problems are ubiquitously characterized by sophisticated computer codes that map parameters or inputs to an underlying physical process. In other situations, experimental setups are used to model the physical process in a laboratory, ensuring high precision while being costly in materials and logistics. In both scenarios, only limited amount of data can be generated by querying the expensive information source at a finite number of inputs or designs. This problem is compounded further in the presence of a high-dimensional input space. State-of-the-art parameter space dimension reduction methods, such as active subspace, aim to identify a subspace of the original input space that is sufficient to explain the output response. These methods are restricted by their reliance on gradient evaluations or copious data, making them inadequate to expensive problems without direct access to gradients. The proposed methodology is gradient-free and fully Bayesian, as it quantifies uncertainty in both the low-dimensional subspace and the surrogate model parameters. This enables a full quantification of epistemic uncertainty and robustness to limited data availability. It is validated on multiple datasets from engineering and science and compared to two other state-of-the-art methods based on four aspects: a) recovery of the active subspace, b) deterministic prediction accuracy, c) probabilistic prediction accuracy, and d) training time. The comparison shows that the proposed method improves the active subspace recovery and predictive accuracy, in both the deterministic and probabilistic sense, when only few model observations are available for training, at the cost of increased training time.
翻译:现代工程问题无处不在,其特点是复杂的计算机代码,这些代码将参数或投入映射到一个基础物理过程。在其他情况下,实验设置被用来在实验室中模拟物理过程,确保高精确度,同时在材料和后勤方面费用昂贵。在两种情况下,只有有限的投入或设计量来查询昂贵的信息源,才能产生数量有限的数据。在高维输入空间存在的情况下,这一问题就更加复杂化了。最先进的参数空间减少方法,如活动子空间,旨在确定原始输入空间的子空间空间的子空间,足以解释产出反应。这些方法受到限制,因为它们依赖梯度评估或可调用的数据,使其不足以解决昂贵的问题,而不能直接接触梯度或设计。提议的方法是无梯度和全巴伊斯式的,因为它能量化了低维度子空间的不确定性和坚固度模型参数。这能够充分量化从工程和科学中得出的多套数据空间,并比其他两种状态精确度数据空间评估。这些方法受到限制,在稳定度、稳定性培训的精确度、稳定度的亚性预测方法以四种方式为基础为基础,只能用来充分量化。