SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 1, Page 62-90, March 2023.
Abstract. The deep active-subspace method is a neural-network based tool for the propagation of uncertainty through computational models with high-dimensional input spaces. Unlike the original active-subspace method, it does not require access to the gradient of the model. It relies on an orthogonal projection matrix constructed with Gram–Schmidt orthogonalization to reduce the input dimensionality. This matrix is incorporated into a neural network as the weight matrix of the first hidden layer (acting as an orthogonal encoder), and optimized using back propagation to identify the active subspace of the input. We propose several theoretical extensions, starting with a new analytic relation for the derivatives of Gram–Schmidt vectors, which are required for back propagation. We also study the use of vector-valued model outputs, which is difficult in the case of the original active-subspace method. Additionally, we investigate an alternative neural network with an encoder without embedded orthonormality, which shows equally good performance compared to the deep active-subspace method. Two epidemiological models are considered as applications, where one requires supercomputer access to generate the training data.

中文翻译：

---

关于深度活动子空间方法

SIAM/ASA 不确定性量化期刊，第 11 卷，第 1 期，第 62-90 页，2023 年 3 月。
抽象的。深度活动子空间方法是一种基于神经网络的工具，用于通过具有高维输入空间的计算模型传播不确定性。与原始的活动子空间方法不同，它不需要访问模型的梯度。它依赖于使用 Gram–Schmidt 正交化构造的正交投影矩阵来降低输入维数。该矩阵作为第一个隐藏层（作为正交编码器）的权重矩阵并入神经网络，并使用反向传播进行优化以识别输入的活动子空间。我们提出了几个理论扩展，从反向传播所需的 Gram-Schmidt 向量导数的新解析关系开始。我们还研究了向量值模型输出的使用，这在原始活动子空间方法的情况下是困难的。此外，我们研究了一种替代神经网络，其编码器没有嵌入正交性，与深度主动子空间方法相比，它表现出同样好的性能。两个流行病学模型被视为应用程序，其中一个需要超级计算机访问来生成训练数据。