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

各种领域的先进技术和科学，如民用基础设施、材料科学和制造，通常可以在数学上表述为设计和控制问题，或者更一般地，作为反转任务。这类任务往往需要以不完整的信息为基础，一方面，根据传感器收集的数据，另一方面，根据可能不完整或依赖于大量未校准参数的数学模型。给出了地下水多孔介质渗流的计算实例，其中数据取自钻孔的压头，模型为含未知参数的压力方程的达西定律：渗透率场。在许多看似不同的应用场景中遇到类似的情况，例如电子阻抗断层扫描，其中人们想要根据位于物体表面的多个电极的电压响应来推断组织内部结构。这些问题中的一个共同挑战是，可用的数据不足以有效地学习潜在的物理过程，并且问题可能具有令人望而却步的巨大计算复杂性。该项目的主要目标是开发可靠的方法来融合数学模型和数据提供的信息，以确保所需的计算复杂性仍然是可承受的，同时所产生的估计器具有高度和可量化的预测能力。为了保证这项工作在广泛的应用中的适用性，将考虑状态和参数估计的足够一般的问题设置。经典的基于模型的方法和来自数据科学的新颖的数据驱动方法之间的相互作用将发挥核心作用。这个项目将让学生和年轻的研究人员对各种相关数学概念和机器学习算法的主要作用有一个明确的定位。这个项目的一个指导主题是寻找贝叶斯逆的替代方案，更加强调确定性的准确性量化，以及揭示内在信息限制的严格的复杂性估计。主要的概念框架是所谓的参数背景数据弱方法，它打开了一个具有以下重要分支的“几何视角”：它基于参数偏微分方程组的稳定变分公式，远远超出了经典的椭圆模型类，通过引用适合问题的非对称弱公式来实现。区分来自泛函和传感器的数据，并将后者提升到适当识别的试验空间，诱导出适合生成最优简化模型的无限维“坐标系”，以及用于回归的机器学习框架，以便仍然尊重内在问题度量。与传统方法不同的是，我们的方法不直接将求逆任务转化为任何先验固定的离散形式。因此，它避免了引入模棱两可的正则化项、剪裁可能重要的尺度信息以及耦合不太兼容的度量。这使人们能够确定反映基本恢复限制的最优基准，并构建满足这些基准或接近于适当的精度-复杂性平衡的估计值。此外，PI将再次在连续水平上使用稳定的弱公式，探索将通常表示为非凸优化问题的(高度不适定的)参数估计与凸优化问题相结合地简化为更良性的状态估计的方法。这揭示了基本的变分公式、解流形的结构和它们通过减基方法或高度非线性的深度神经网络的可逼近性之间的相互作用。这项研究将导致严格的复杂性和准确性量化，并减少特别和模糊问题截断的需要。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.