State and Parameter Estimation: Variationally Stable Models and Physics-Informed Learning

状态和参数估计：变分稳定模型和物理知情学习

基本信息

批准号：
2012469
负责人：
Wolfgang Dahmen
金额：
$ 22.46万
依托单位：
University of South Carolina at Columbia
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012469&HistoricalAwards=false
关键词：
State Parameter Estimation Variationally Stable

项目摘要

Advancing technology and science in a variety of areas, such as civil infrastructure, material science, and manufacturing, can often be formulated mathematically as design and control problems, or more generally, as inversion tasks. Such tasks often need to be based on incomplete information given, on the one hand, in terms of data collected by sensors, and on the other hand, in terms of a mathematical model which may be incomplete or depend on a large number of uncalibrated parameters. An illustrative example concerns the estimation of groundwater porous media flow where the data are pressure heads taken from boreholes and the model is Darcy's law for the pressure equation with an unknown parameter: permeability field. A similar situation is encountered in many seemingly different application scenarios such as Electron Impedance Tomography where one wants to infer inner tissue structure from voltage responses at a number of electrodes, located at the surface of the object. A common challenge in these problems is that the available data are not sufficient to effectively learn the underlying physical process, and that the problem may have a prohibitively large computational complexity. The key objective of this project is to develop robust methods for fusing the information provided by the mathematical model and by the data so as to ensure that the required computational complexity remains affordable while the resulting estimators have a high and quantifiable predictive capability. To warrant the applicability of the work to a broad range of applications, a sufficiently general problem setting for state and parameter estimation will be considered. A central role will be played by the interplay between classical model-based approaches and novel data-driven methodologies from data science. This project will give students and young researchers a clear orientation on the principal role of a variety of relevant mathematical concepts and machine learning algorithms.A guiding theme in this project is the search for alternatives to Bayesian inversion with a stronger emphasis on deterministic accuracy quantification with rigorous complexity estimates revealing intrinsic information limits. The main conceptual framework is the so called Parametrized-Background Data-Weak method, which opens a “geometric perspective” with the following important ramifications: it is based on stable variational formulations for the parametric partial differential equations, well beyond the classical elliptic model classes, by invoking suitable problem-adapted nonsymmetric weak formulations. Distinguishing data from the functionals and sensors, and lifting the latter to the properly identified trial space, induces an infinite-dimensional “coordinate system” that accommodates the generation of optimal reduced models as well as a machine learning framework for regression so as to still respect intrinsic problem metrics. Different from the conventional approaches, our method does not cast the inversion task directly into any a priori fixed discrete form. Thus, it avoids introducing ambiguous regularization terms, clipping possibly important scale information and coupling less compatible metrics. This allows one to identify optimality benchmarks reflecting essential recovery limitations and construct estimators that meet these benchmarks or come close within a proper accuracy-complexity balance. Moreover, using again stable weak formulations on a continuous level, the PIs will explore ways of reducing (highly ill-posed) parameter estimation typically formulated as a non-convex optimization problem to more benign state estimation in combination with a convex optimization problem. This sheds light on the interplay between the underlying variational formulations, structure of solution manifolds, and their approximability by reduced basis methods or highly nonlinear deep neural networks. This research will lead to rigorous complexity and accuracy quantification, and reduce the need for ad hoc and ambiguous problem truncations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在各个领域（例如民用基础设施，材料科学和制造业）中推进技术和科学，通常可以数学上以设计和控制问题为数学，或更普遍地作为倒置任务。此类任务通常需要基于一方面，就传感器收集的数据而言，就数学模型而言，该信息可能是不完整或取决于大量未校准的参数而言。说明性示例涉及地下水多孔培养基流的估计，其中数据是从钻孔中取的压力头，并且模型是带有未知参数的压力方程的达西定律：渗透率。在许多看似不同的应用程序场景（例如电子阻抗断层扫描）中也遇到了类似的情况，在这些情况下，人们想从位于表面上的许多电子中的电压响应来内部组织结构，在这些问题上是一个共同的挑战，即可用的数据不足以有效地学习潜在的物理过程，并且问题可能具有合理的大型计算复杂性。该项目的主要目的是开发可靠的方法来融合数学模型提供的信息和数据，以确保所需的计算复杂性保持负担得起，而所得估计器具有高且可量化的预测能力。为了保证工作适用于广泛的应用程序，将考虑对状态和参数估计的足够一般的问题设置。基于经典模型的方法与数据科学的新型数据驱动方法之间的相互作用将扮演核心角色。该项目将使学生和年轻的研究人员明确地取向了各种相关数学概念和机器学习算法的主要作用。该项目的指导主题是寻找贝叶斯倒置的替代方案，并更加强调强调确定严格的复杂性估计的精确性量化，从而估算了揭示内在信息的限制。主要的概念框架是所谓的参数 - 背景数据吸引方法，它通过以下重要分析打开了“几何透视”：它基于参数部分偏微分方程的稳定变异公式，远远超出了经典椭圆模型类别，通过调用合适的问题适当的问题适合问题的非对称的非对称的非对称的非对称非对称的弱弱的形式。将数据与功能和传感器区分开，并将较晚的数据提升到正确识别的试验空间，引起无限维度的“坐标系”，可容纳最佳减少模型的生成以及用于回归的机器学习框架，以便仍然尊重内在问题指标。与常规方法不同，我们的方法不会直接将反转任务直接投入到任何先前的固定离散表中。这样一来，它避免引入模棱两可的术语，剪辑可能的重要规模信息并耦合较少的兼容指标。这允许人们确定反映基本恢复限制和构造符合这些基准测试或接近适当准确性复杂性平衡的最佳基准。此外，在连续水平上再次使用稳定的弱公式，PI将探索减少（高度不良的）参数估计的方法，通常以非凸优化问题为单位，以与凸优化问题结合使用更良性的状态估计。这阐明了潜在的变化公式，解决方案歧管的结构以及通过降低基础方法或高度非线性深神经网络之间的相互作用。这项研究将导致严格的复杂性和准确性量化，并减少对临时和模棱两可的问题截断的需求。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准来诚实地认为通过评估来诚实地支持。