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，能够完成多项任务，如复杂物理现象的建模、关键属性的预测以及设计优化。这些计算机模型的高保真度使其在优化过程中需要进行数百次查询，计算成本高昂，因此通常依赖简化模型，但代价是牺牲预测准确性和精度。为此，数据驱动的替代建模方法在模拟昂贵计算机模型的行为方面展现出巨大潜力。然而，这类方法的主要瓶颈在于难以处理高维输入，且需要相对较大的数据集。在此类问题中，感兴趣的输入和输出量为高维张量。常用的替代建模方法需要大量计算评估，这阻碍了其他数值任务（如不确定性量化和统计分析）的开展。在本研究中，我们提出了一种端到端方法，将高维图像状输入映射至高维输出或其关键统计量。我们的方法采用两个主要框架，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) 在低维空间中建模输入-输出关系；(c) 将领域特定的物理约束作为掩码融入其中。为实现输入降维，我们利用主成分分析，并结合两种替代建模方法：(a) 贝叶斯混合建模，以及(b) DeepHyper的深度神经网络。我们在线性弹性应力场数据问题上验证了该方法的适用性。