Computational and machine learning methods for model reduction, uncertainty propagation, and parameter identification in fluid and solid mechanics

工程建模中的大多数流体和固体问题都是用时变和参数化的非线性偏微分方程来描述的。传统计算力学方法的分辨率可能过于昂贵，特别是在不确定性量化或优化预测的背景下，无法实现快速预测。我们建议研究先进的机器学习方法，旨在通过降维替代模型来表示高保真计算模型。此外，还将研究不确定度量化的不同方法。的确，对自然和工业过程的可靠建模预测应包括对各种来源可能产生的不确定性的量化。该计划还支持对并行计算的持续兴趣，以实现大规模模拟的快速预测。虽然这些拟议的活动旨在开发新的方法和算法，但研究计划也是应用驱动的。我们将重点关注两个具有挑战性的工程应用，这是由实际需求驱动的。然而，所提出的方法将不限于这些应用。-使用机器学习的概率洪水地图：洪水是最昂贵的自然灾害之一。因此，政府和机构需要制定可靠和准确的洪水危险区地图，作为其预防措施的一部分。这里的主要目标是开发预测工具，将先进的计算方法与机器学习相结合，以建立具有概率信息的精确地图；并且，在极端紧急事件的情况下，允许快速预测，几乎是实时的。-物理参数识别：复杂系统建模中使用的本构参数通常具有高度的不确定性。逆分析提供了一种识别这些参数的方法。研究了选择性激光熔化（SLM）增材制造过程中众多参数的辨识和优化问题。增材制造是一项现代技术，已被广泛应用于汽车、航空航天和医疗等多个行业。预计该研究项目将有助于提高数值建模、机器学习和参数优化等几个方面的知识，并增加对水力学和增材制造不确定性的分析。培养的高素质人才（博士5名，博士后1名，硕士2名，本科生5名）将有利于整个社会和加拿大的相关行业。许多创新出版物和与工业伙伴的合作具有很大的潜力。这项研究还将导致高性能代码的开发，这些代码可以用于更多的研究和应用项目。

Most fluid and solid problems in engineering modeling are described by time-dependent and parametrized nonlinear partial differential equations. Their resolution with traditional computational mechanics methods may be too expensive, especially in the context of predictions with uncertainty quantification or optimization, to allow for rapid predictions. We propose to investigate advanced machine learning methods aimed at representing high-fidelity computational models by means of reduced-dimension surrogate ones. In addition, different approaches will be studied for the uncertainty quantification. Indeed, reliable modeling predictions of natural and industrial processes should include quantification of the uncertainties that may arise from various sources. This program also supports a continuous interest in parallel computing to enable rapid predictions for large-scale simulations. While these proposed activities are geared towards developing new approaches and algorithms, the research program is also application-driven. We will focus on two challenging engineering applications, motivated by real needs. However, the proposed methods  will not be restricted to these applications. - Probabilistic flooding maps using machine learning: Floods are among the costliest natural disasters. Governments and agencies are therefore required to develop reliable and accurate maps of flood risk areas as part of their preventive measures. The main objective here is to develop predictive tools that combine advanced computational methods with machine learning to establish accurate maps with probabilistic information; and, in the case of an extreme emergency event, to allow rapid predictions, almost in real-time. - Physical parameter identification: The constitutive parameters used in the modeling of complex systems are often associated with high degrees of uncertainties. Inverse analysis provides a way to identify these parameters. We consider the identification and optimization of the numerous parameters involved in the selective laser melting (SLM) additive manufacturing process. Additive manufacturing is a modern technology that has been used across a diverse range of industries, including automotive, aerospace and medical, among others. It is anticipated that this research program will contribute to the advancement of knowledge about several aspects of numerical modeling, machine learning and parameter optimization, and add to the analysis of uncertainties in hydraulics and additive manufacturing. The training of highly qualified personnel (5 Ph.D.s, one postdoctoral fellow, 2 M.SC students and 5 undergraduate students) will be beneficial to society as a whole, as well as to relevent industries in Canada. There is a great potential for a number of innovative publications and collaborations with industrial partners. This research will also result in the development of high-performance codes that can be used for even more research and applied projects.