We present a distributed active subspace method for training surrogate models of complex physical processes with high-dimensional inputs and function valued outputs. Specifically, we represent the model output with a truncated Karhunen–Loève (KL) expansion, screen the structure of the input space with respect to each KL mode via the active subspace method, and finally form an overall surrogate model of the output by combining surrogates of individual output KL modes. To ensure scalable computation of the gradients of the output KL modes, needed in active subspace discovery, we rely on adjoint-based gradient computation. The proposed method combines benefits of active subspace methods for input dimension reduction and KL expansions used for spectral representation of the output field. We provide a mathematical framework for the proposed method and conduct an error analysis of the mixed KL active subspace approach. Specifically, we provide an error estimate that quantifies errors due to active subspace projection and truncated KL expansion of the output. We demonstrate the numerical performance of the surrogate modeling approach with an application example from biotransport.

我们提出了一种分布式主动子空间方法，用于训练具有高维输入和函数值输出的复杂物理过程的替代模型。具体来说，我们用截短的Karhunen-Loève（KL）展开表示模型输出，通过活动子空间方法针对每个KL模式筛选输入空间的结构，最后通过组合替代来形成输出的整体替代模型单个输出KL模式的设置。为了确保活动子空间发现所需的可扩展输出KL模式梯度的计算，我们依赖于基于伴随的梯度计算。所提出的方法结合了用于输入尺寸减小的有源子空间方法的优势以及用于输出场频谱表示的KL扩展。我们为提出的方法提供了数学框架，并对混合KL主动子空间方法进行了误差分析。具体来说，我们提供了一个误差估算，该误差估算可量化由于活动子空间投影和输出的KL截短而导致的误差。我们用生物运输的一个应用实例证明了替代建模方法的数值性能。