Stress and material deformation field predictions are among the most important tasks in computational mechanics. These predictions are typically made by solving the governing equations of continuum mechanics using finite element analysis, which can become computationally prohibitive considering complex microstructures and material behaviors. Machine learning (ML) methods offer potentially cost effective surrogates for these applications. However, existing ML surrogates are either limited to low-dimensional problems and/or do not provide uncertainty estimates in the predictions. This work proposes an ML surrogate framework for stress field prediction and uncertainty quantification for diverse materials microstructures. A modified Bayesian U-net architecture is employed to provide a data-driven image-to-image mapping from initial microstructure to stress field with prediction (epistemic) uncertainty estimates. The Bayesian posterior distributions for the U-net parameters are estimated using three state-of-the-art inference algorithms: the posterior sampling-based Hamiltonian Monte Carlo method and two variational approaches, the Monte-Carlo Dropout method and the Bayes by Backprop algorithm. A systematic comparison of the predictive accuracy and uncertainty estimates for these methods is performed for a fiber reinforced composite material and polycrystalline microstructure application. It is shown that the proposed methods yield predictions of high accuracy compared to the FEA solution, while uncertainty estimates depend on the inference approach. Generally, the Hamiltonian Monte Carlo and Bayes by Backprop methods provide consistent uncertainty estimates. Uncertainty estimates from Monte Carlo Dropout, on the other hand, are more difficult to interpret and depend strongly on the method's design.

翻译：应力场与材料变形场预测是计算力学中最为重要的任务之一。这些预测通常通过有限元分析求解连续介质力学控制方程来实现，但考虑到复杂的微观结构和材料行为，其计算成本可能变得难以承受。机器学习方法为此类应用提供了潜在的高性价比替代方案。然而，现有的机器学习替代模型要么局限于低维问题，要么无法提供预测中的不确定性估计。本研究提出了一种适用于多种材料微观结构的应力场预测与不确定性量化的机器学习替代框架。采用改进的贝叶斯U-net架构，通过数据驱动的图像到图像映射，从初始微观结构到应力场进行预测，并提供预测（认知）不确定性估计。U-net参数的贝叶斯后验分布使用三种先进推理算法进行估计：基于后验采样的哈密顿蒙特卡洛方法，以及两种变分方法——蒙特卡洛Dropout方法和反向传播贝叶斯算法。针对纤维增强复合材料和多晶微观结构应用，系统比较了这些方法的预测精度与不确定性估计结果。研究表明，与有限元分析解相比，所提方法能获得高精度预测，而不确定性估计则取决于推理方法。总体而言，哈密顿蒙特卡洛和反向传播贝叶斯方法能提供一致的不确定性估计。相比之下，蒙特卡洛Dropout方法的不确定性估计更难以解释，且高度依赖于方法的设计参数。