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.

中文翻译：

---

用于预测全场材料响应不确定性的贝叶斯神经网络


应力和材料变形场预测是计算力学中最重要的任务之一。这些预测通常是通过使用有限元分析求解连续介质力学的控制方程来做出的，考虑到复杂的微观结构和材料行为，这可能会在计算上变得令人望而却步。机器学习 （ML） 方法为这些应用程序提供了可能具有成本效益的替代方法。然而，现有的 ML 代理要么局限于低维问题，要么在预测中不提供不确定性估计。这项工作提出了一个 ML 替代框架，用于不同材料微观结构的应力场预测和不确定性量化。采用改进的贝叶斯 U-net 架构提供从初始微观结构到应力场的数据驱动的图像到图像映射，并带有预测（认识）不确定性估计。使用三种最先进的推理算法估计 U-net 参数的贝叶斯后验分布：基于后验采样的哈密顿蒙特卡洛方法和两种变分方法，即蒙特卡洛 Dropout 方法和反向传播的贝叶斯算法。针对纤维增强复合材料和多晶微结构应用，对这些方法的预测准确性和不确定性估计进行了系统比较。结果表明，与 FEA 解决方案相比，所提出的方法产生了高精度的预测，而不确定性估计取决于推理方法。通常，通过 Backprop 方法的哈密顿蒙特卡洛和贝叶斯方法提供一致的不确定性估计。 另一方面，Monte Carlo Dropout 的不确定性估计更难解释，并且在很大程度上取决于方法的设计。