Modern computational methods, involving highly sophisticated mathematical formulations, enable several tasks like modeling complex physical phenomenon, predicting key properties and design optimization. The higher fidelity in these computer models makes it computationally intensive to query them hundreds of times for optimization and one usually relies on a simplified model albeit at the cost of losing predictive accuracy and precision. Towards this, data-driven surrogate modeling methods have shown a lot of promise in emulating the behavior of the expensive computer models. However, a major bottleneck in such methods is the inability to deal with high input dimensionality and the need for relatively large datasets. With such problems, the input and output quantity of interest are tensors of high dimensionality. Commonly used surrogate modeling methods for such problems, suffer from requirements like high number of computational evaluations that precludes one from performing other numerical tasks like uncertainty quantification and statistical analysis. In this work, we propose an end-to-end approach that maps a high-dimensional image like input to an output of high dimensionality or its key statistics. Our approach uses two main framework that perform three steps: a) reduce the input and output from a high-dimensional space to a reduced or low-dimensional space, b) model the input-output relationship in the low-dimensional space, and c) enable the incorporation of domain-specific physical constraints as masks. In order to accomplish the task of reducing input dimensionality we leverage principal component analysis, that is coupled with two surrogate modeling methods namely: a) Bayesian hybrid modeling, and b) DeepHyper's deep neural networks. We demonstrate the applicability of the approach on a problem of a linear elastic stress field data.

翻译：现代计算方法涉及高度复杂的数学公式，能够完成诸如复杂物理现象建模、关键属性预测和设计优化等任务。这些计算机模型的高保真度使得对其进行数百次优化查询的计算量巨大，因此研究人员通常依赖于简化模型，但这样会牺牲预测准确性和精度。为此，数据驱动的代理建模方法在模拟昂贵计算机模型行为方面展现出巨大潜力。然而，这类方法的主要瓶颈在于无法处理高维输入问题以及需要相对较大的数据集。对于此类问题，输入和输出感兴趣量均为高维张量。常用的代理建模方法面临需要大量计算评估的约束，从而难以执行不确定性量化和统计分析等其他数值任务。本研究提出了一种端到端方法，将高维图像类输入映射为高维输出或其关键统计量。该方法采用两大框架执行三个步骤：(a) 将输入和输出从高维空间降至低维空间，(b) 在低维空间中建立输入-输出关系模型，(c) 实现领域特定物理约束作为掩码的整合。为完成输入降维任务，我们利用主成分分析，并将其与两种代理建模方法耦合：一是贝叶斯混合建模，二是DeepHyper深度神经网络。我们在线弹性应力场数据问题上验证了该方法的适用性。