Computational Methods for Verification and Validation with Applications in Mechanics

计算科学与工程是一门以物理现象和工程系统的计算机模拟为重点的学科，多年来已经发展得非常成熟，目前已被广泛考虑用于设计和决策，无论是材料科学、航空航天应用、与天气有关的灾害监测、制造过程的控制等。然而，人们也承认，由于对数学模型及其参数的不完善了解，人们对计算机预测的信心往往是有限的。事实上，由于被模拟的多物理和多尺度系统的复杂性，确保预测在数量上可靠比以往任何时候都更加重要。计算模型需要经过广泛的验证和确认过程，以评估模拟结果的可信度。代码和解决方案验证是确定计算模型产生的数值结果是否具有足够的精度，以便与实验数据进行有意义的比较的过程。验证是确定数学模型可以预测与必须做出的决定有关的物理事件的准确性的过程。核查和验证的主题并不新鲜，但目前可用的办法和方法仍然更多地属于一门艺术，特别是在验证的情况下，而不是科学方法。该研究计划旨在开发用于预测建模的新理论、新方法和新算法，并将集中于以下目标：1)构建自适应且经过验证的降阶模型(低阶近似、适当的正交分解等)。对于感兴趣的具体数量，目标是获得替代模型，以快速和准确地统计表征感兴趣的输出量；2)在数据稀缺的情况下，使用机器学习方法来增强用于参数推理和模型验证的数据集的信息含量(例如，神经网络、深度学习、模型选择、交叉验证、重采样方法等)；3)应用图形方法来评估验证场景和数据的选择的相关性，以便适当地使用模型。在该项目中开发的方法将在几个具有工程意义的应用中进行测试。该项目的成果将是一系列用于验证和确认计算模型的计算工具。这项工作的原创性将是开发为计算感兴趣的数量而量身定做的降阶模型，并整合机器学习技术以提高验证过程的可靠性。该项目不仅应该在计算力学领域产生影响，而且还应该在严重依赖复杂物理现象的计算机模拟的任何其他工程应用中产生影响。

Computational Sciences and Engineering, the discipline which focuses on computer simulations of physical phenomena and engineering systems, has gained great maturity over the years to the point that it is today widely considered for design and decision-making, either for material sciences, aerospace applications, weather-related disaster monitoring, control of manufacturing processes, etc. Yet, it is also acknowledged that confidence in computer predictions is often limited because of the imperfect knowledge of the mathematical models and their parameters. In fact, due to the complexity of the multiphysics and multiscale systems being simulated, it is, more than ever, essential to ensure that predictions be quantitatively reliable. The computational models need to be extensively subjected to verification and validation processes in order to assess the credibility of the simulation results. Code and Solution Verification is the process of determining whether computational models produce numerical results with sufficient accuracy for meaningful comparisons with experimental data. Validation is the process of determining the accuracy by which mathematical models can predict physical events with respect to a decision that has to be made. The topics of verification and validation are not new, but approaches and methodologies currently available still pertain more to an art, especially in the case of validation, than to a scientific approach. The proposed research program aims at developing new theories, methodologies, and algorithms for predictive modeling, and will concentrate on the following objectives: 1) Construction of adaptive and verified reduced-order models (low rank approximations, proper orthogonal decomposition, etc.) with respect to specific quantities of interest, the goal being to obtain surrogate models for fast and accurate statistical characterization of output quantities of interest; 2) Use of machine learning methods to enhance the information content of datasets for parameter inference and model validation in the case of scarce data (e.g. neural networks, deep learning, model selection, cross-validation, resampling methods, etc.); 3) Application of graphical methods to assess the relevance in the choice of validation scenarios and data to properly exercise a model. The methodologies developed in the project will be tested on several applications of engineering interest. Outcomes of the project will be a collection of computational tools for verification and validation of computational models. The originality of the work will be the development of reduced-order models tailored to the calculation of quantities of interest and the integration of machine learning techniques to enhance the reliability of validation processes. The project should have an impact not only in the field of computational mechanics but also in any other engineering applications that heavily rely on computer simulations of complex physical phenomena.