Guo Yulin, Nath Paromita, Mahadevan Sankaran, Witherell Paul
Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235 USA.
Department of Mechanical Engineering, Rowan University, Glassboro, NJ 08028 USA.
Struct Multidiscipl Optim. 2024;67(7):122. doi: 10.1007/s00158-024-03816-9. Epub 2024 Jul 10.
This paper investigates a novel approach to efficiently construct and improve surrogate models in problems with high-dimensional input and output. In this approach, the principal components and corresponding features of the high-dimensional output are first identified. For each feature, the active subspace technique is used to identify a corresponding low-dimensional subspace of the input domain; then a surrogate model is built for each feature in its corresponding active subspace. A low-dimensional adaptive learning strategy is proposed to identify training samples to improve the surrogate model. In contrast to existing adaptive learning methods that focus on a scalar output or a small number of outputs, this paper addresses adaptive learning with high-dimensional input and output, with a novel learning function that balances exploration and exploitation, i.e., considering unexplored regions and high-error regions, respectively. The adaptive learning is in terms of the active variables in the low-dimensional space, and the newly added training samples can be easily mapped back to the original space for running the expensive physics model. The proposed method is demonstrated for the numerical simulation of an additive manufacturing part, with a high-dimensional field output quantity of interest (residual stress) in the component that has spatial variability due to the stochastic nature of multiple input variables (including process variables and material properties). Various factors in the adaptive learning process are investigated, including the number of training samples, range and distribution of the adaptive training samples, contributions of various errors, and the importance of exploration versus exploitation in the learning function.
本文研究了一种在高维输入和输出问题中高效构建和改进代理模型的新方法。在这种方法中，首先识别高维输出的主成分和相应特征。对于每个特征，使用主动子空间技术识别输入域的相应低维子空间；然后在其相应的主动子空间中为每个特征构建一个代理模型。提出了一种低维自适应学习策略来识别训练样本以改进代理模型。与现有专注于标量输出或少量输出的自适应学习方法不同，本文处理具有高维输入和输出的自适应学习，具有一种平衡探索和利用的新颖学习函数，即分别考虑未探索区域和高误差区域。自适应学习是针对低维空间中的主动变量进行的，新添加的训练样本可以很容易地映射回原始空间以运行昂贵的物理模型。针对增材制造零件的数值模拟展示了所提出的方法，该零件具有由于多个输入变量（包括工艺变量和材料特性）的随机性而在空间上具有变化性的高维场输出感兴趣量（残余应力）。研究了自适应学习过程中的各种因素，包括训练样本数量、自适应训练样本的范围和分布、各种误差的贡献以及学习函数中探索与利用的重要性。
