Benjamin Sanderse

The application of operator inference to parametric problems whose operators exhibit affine parameter dependency results in a minimization problem with two issues. First, the solution to the minimization problem is generally not unique, so we might obtain a different solution than the one we are interested in. Second, the large dimensions of the minimization problem can cause prohibitively high computational costs. In this work, we resolve these issues by extending previous work on exact operator inference to the parametric case. Concretely, we propose a snapshot data generation method that guarantees the uniqueness of the solution and thereby the exact reconstruction of the corresponding intrusive operators of projection-based reduced order models. Furthermore, we show that the monolithic problem that comprises the snapshot data of all sampled parameters can be decoupled into separate problems for only one parameter sample each. This decoupling results in smaller subproblems that are embarrassingly parallel, numerically more stable and can be solved more efficiently. In numerical experiments, we demonstrate this decoupling for the heat equation with two different temperature-, parameter- and space-dependent thermal conductivities.

Closure problems are omnipresent when simulating multiscale systems, where some quantities and processes cannot be fully prescribed despite their effects on the simulation's accuracy. Recently, scientific machine learning approaches have been proposed as a way to tackle the closure problem, combining traditional (physics-based) modeling with data-driven (machine-learned) techniques, typically through enriching differential equations with neural networks. This paper reviews the different reduced model forms, distinguished by the degree to which they include known physics, and the different objectives of a priori and a posteriori learning. The importance of adhering to physical laws (such as symmetries and conservation laws) in choosing the reduced model form and choosing the learning method is discussed. The effect of spatial and temporal discretization and recent trends toward discretization-invariant models are reviewed. In addition, we make the connections between closure problems and several other research disciplines: inverse problems, Mori-Zwanzig theory, and multi-fidelity methods. In conclusion, much progress has been made with scientific machine learning approaches for solving closure problems, but many challenges remain. In particular, the generalizability and interpretability of learned models is a major issue that needs to be addressed further.