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

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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.