Scalable Probabilistic Modeling and Machine Learning With Dimensionality Reduction for Expensive High-Dimensional Problems

Abstract

Abstract Modern computational methods involving highly sophisticated mathematical formulations enable several tasks like modeling complex physical phenomena, predicting key properties, and optimizing design. The higher fidelity in these computer models makes it computationally intensive to query them hundreds of times for optimization. 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 much promise in emulating the behavior of 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. In certain cases, the high dimensionality of the input space can be attributed to its image-like characteristics, for example, the stress and displacement fields of continuums. 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 many computational evaluations that precludes one from performing other numerical tasks like uncertainty quantification and statistical analysis. This work proposes 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 frameworks 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. To reduce input dimensionality, we leverage principal component analysis, coupled with two surrogate modeling methods: a) Bayesian hybrid modeling and b) DeepHyper’s deep neural networks. We demonstrate the approach’s applicability to a linear elastic stress field data problem. We perform numerical studies to study the effect of the two end-to-end workflows and the effect of data size. Key insights and conclusions are provided, which can aid such efforts in surrogate modeling and engineering optimization.