CHARLEMAGNE DISTINGUISHED LECTURE SERIES - Willcox Seminar
Prof. Karen Willcox, Ph.D. - Projection-based Model Reduction: Formulations for Scientific Machine Learning
The field of model reduction encompasses a broad range of methods that seek efficient low-dimensional representations of an underlying high-fidelity model. A large class of model reduction methods are projection-based; that is, they derive the low-dimensional approximation by projection of the original large-scale model onto a low-dimensional subspace. Model reduction has clear connections to machine learning. The difference in fields is perhaps largely one of history and perspective: model reduction methods have grown from the scientific computing community, with a focus on reducing high-dimensional models that arise from physics-based modeling, whereas machine learning has grown from the computer science community, with a focus on creating low-dimensional models from black-box data streams. This talk will describe two methods that blend the two perspectives and provide advances towards achieving the goals of Scientific Machine Learning. The first method combines lifting--the introduction of auxiliary variables to transform a general nonlinear model to a model with polynomial nonlinearities--with proper orthogonal decomposition (POD). The result is a data-driven formulation to learn the low-dimensional model directly from data, but a key aspect of the approach is that the lifted state-space in which the learning is achieved is derived using the problem physics. The second method combines a low-dimensional POD parametrization of quantities of interest with machine learning methods to learn the map between the input parameters and the POD expansion coefficients. The use of particular solutions in the POD expansion provides a way to embed physical constraints, such as boundary conditions. Case studies demonstrate the importance of embedding physical constraints within learned models, and also highlight the important point that the amount of model training data available in an engineering setting is often much less than it is in other machine learning applications, making it essential to incorporate knowledge from physical models.