Many real-world applications demand accurate and fast predictions, as well as reliable uncertainty estimates. However, quantifying uncertainty on high-dimensional predictions is still a severely under-invested problem, especially when input-output relationships are non-linear. To handle this problem, the present work introduces an innovative approach that combines autoencoder deep neural networks with the probabilistic regression capabilities of Gaussian processes. The autoencoder provides a low-dimensional representation of the solution space, while the Gaussian process is a Bayesian method that provides a probabilistic mapping between the low-dimensional inputs and outputs. We validate the proposed framework for its application to surrogate modeling of non-linear finite element simulations. Our findings highlight that the proposed framework is computationally efficient as well as accurate in predicting non-linear deformations of solid bodies subjected to external forces, all the while providing insightful uncertainty assessments.

翻译：许多实际应用需要准确快速的预测以及可靠的不确定性估计。然而，高维预测中的不确定性量化仍是一个研究严重不足的问题，特别是在输入输出关系呈现非线性的情况下。为解决这一问题，本研究提出了一种创新方法，将自编码器深度神经网络与高斯过程的概率回归能力相结合。自编码器提供了解空间的低维表示，而高斯过程作为一种贝叶斯方法，能够建立低维输入与输出之间的概率映射。我们通过非线性有限元模拟的代理建模应用验证了所提框架的有效性。研究结果表明，该框架在计算效率与预测固体受外力作用下的非线性变形精度方面均表现优异，同时能够提供具有洞察力的不确定性评估。