{"title":"Bayesian neural networks for predicting uncertainty in full-field material response","authors":"","doi":"10.1016/j.cma.2024.117486","DOIUrl":null,"url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":null,"pages":null},"PeriodicalIF":6.9000,"publicationDate":"2024-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computer Methods in Applied Mechanics and Engineering","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0045782524007400","RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
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.
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实现的,考虑到复杂的微结构和材料行为,这种方法在计算上会变得非常昂贵。机器学习(ML)方法为这些应用提供了具有潜在成本效益的替代方法。然而,现有的 ML 替代方法要么局限于低维问题,要么不能提供预测的不确定性估计。本研究提出了一种 ML 代理框架,用于应力场预测和各种材料微结构的不确定性量化。采用了一种改进的贝叶斯 U 型网络结构,提供从初始微观结构到应力场的数据驱动图像到图像映射,并提供预测(认识)不确定性估计。U 网参数的贝叶斯后验分布是利用三种最先进的推理算法估算的:基于后验采样的哈密尔顿蒙特卡洛方法和两种变异方法,即蒙特卡洛剔除法和贝叶斯后验算法。针对纤维增强复合材料和多晶微结构应用,对这些方法的预测精度和不确定性估计进行了系统比较。结果表明,与有限元分析解决方案相比,所提出的方法可获得较高的预测精度,而不确定性估计则取决于推理方法。一般来说,汉密尔顿蒙特卡洛法和贝叶斯反推法提供的不确定性估计值是一致的。而蒙特卡洛剔除法的不确定性估计值则更难解释,并且在很大程度上取决于方法的设计。
期刊介绍:
Computer Methods in Applied Mechanics and Engineering stands as a cornerstone in the realm of computational science and engineering. With a history spanning over five decades, the journal has been a key platform for disseminating papers on advanced mathematical modeling and numerical solutions. Interdisciplinary in nature, these contributions encompass mechanics, mathematics, computer science, and various scientific disciplines. The journal welcomes a broad range of computational methods addressing the simulation, analysis, and design of complex physical problems, making it a vital resource for researchers in the field.