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

在民用基础设施、材料科学和制造业等各个领域推进技术和科学，通常可以用数学形式表述为设计和控制问题，或者更一般地说，作为反演任务。这类任务往往需要基于所提供的不完整信息，一方面是传感器收集的数据，另一方面是可能不完整或依赖于大量未校准参数的数学模型。一个说明性的例子涉及地下水多孔介质流动的估计，其中数据是从钻孔中获取的压力头，模型是带有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也会遇到类似的情况，例如电子阻抗断层扫描，人们希望通过位于物体表面的许多电极的电压响应来推断内部组织结构。在这些问题中，一个共同的挑战是可用的数据不足以有效地了解底层的物理过程，并且问题可能具有令人望而却步的大计算复杂性。这个项目的主要目标是开发健壮的方法来融合由数学模型和数据提供的信息，以确保所需的计算复杂性保持在可承受的范围内，同时产生的估计器具有高的和可量化的预测能力。为了保证工作适用于广泛的应用，将考虑一个足够普遍的状态和参数估计问题设置。经典的基于模型的方法和来自数据科学的新型数据驱动方法之间的相互作用将发挥核心作用。该项目将为学生和年轻研究人员提供一个明确的方向，了解各种相关数学概念和机器学习算法的主要作用。这个项目的一个指导主题是寻找贝叶斯反演的替代方案，更强调确定性的精度量化，严格的复杂性估计揭示了内在的信息限制。主要的概念框架是所谓的参数化背景数据弱方法，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的稳定变分公式，远远超出了经典的椭圆模型类，通过调用合适的问题适应的非对称弱公式。将数据从函数和传感器中区分出来，并将后者提升到适当识别的试验空间，可以产生一个无限维的“坐标系统”，该“坐标系统”可以容纳最优简化模型的生成，以及用于回归的机器学习框架，从而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反演任务转换为任何先验的固定离散形式。因此，它避免引入模棱两可的正则化术语，剪裁可能重要的尺度信息和耦合不太兼容的度量。这允许识别反映基本恢复限制的最优性基准，并构建满足这些基准或接近适当的准确性-复杂性平衡的估计器。此外，在连续水平上再次使用稳定的弱公式，pi将探索将通常作为非凸优化问题表述的（高度不适定的）参数估计减少为与凸优化问题相结合的更良性的状态估计的方法。这揭示了潜在的变分公式之间的相互作用，解流形的结构，以及它们的近似性通过减少基方法或高度非线性深度神经网络。该研究将导致严格的复杂性和准确性量化，并减少对特殊和模糊问题截断的需求。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.