'Role of Physics in Physics-Informed Machine Learning' by Prof. Daniel Tartakovsky

Machine learning tools for equation discovery require large amounts of data that are typically computer generated rather than experimentally observed. Multiscale modeling and stochastic simulations are two areas where learning on simulated data can lead to such discovery. In both, the data are generated with a reliable but impractical model, e.g., molecular dynamics simulations, while a model on the scale of interest is uncertain, requiring phenomenological constitutive relations and ad-hoc approximations. We replace the human discovery of such models, which typically involves spatial/stochastic averaging or coarse-graining, with a lasso-based machine-learning strategy that can be executed in two modes. The first, direct equation-learning, discovers a differential operator from the whole dictionary. The second, constrained equation-learning, discovers only those terms in the differential operator that need to be discovered, i.e., learns closure approximations. In either case, physical systems are characterized by inherent symmetries that exist within the units of the variables that quantify their behavior. These symmetries enable a lossless dimensionality reduction through a clever change of variables that transforms the solution space into one without physical dimensions, as proposed by the Buckingham theorem. A series of examples demonstrates the accuracy, robustness, and limitations of our approach to equation discovery.