Parsimony as the ultimate regularizer for physics-informed machine learning

Data-driven modeling continues to be enabled by modern machine learning algorithms and deep learning architectures. The goals of such efforts revolve around the generation of models for prediction, characterization, and control of complex systems. In the context of physics and engineering, extrapolation and generalization are critical aspects of model discovery that are empowered by various aspects of parsimony. Parsimony can be encoded (i) in a low-dimensional coordinate system, (ii) in the representation of governing equations, or (iii) in the representation of parametric dependencies. In what follows, we illustrate techniques that leverage parsimony in deep learning to build physics-based models, culminating in a deep learning architecture that is parsimonious in coordinates and also in representing the dynamics and their parametric dependence through a simple normal form. Ultimately, we argue that promoting parsimony in machine learning results in more physical models, i.e., models that generalize and are parametrically represented by governing equations.