Physics-Informed Machine Learning and Dynamical Systems

Abstract:

Modeling dynamical systems from observations of their states and flows is a fundamental challenge in science and engineering. This modeling can be done numerically, by training a machine learning model to regress the system's governing equation, or analytically, by constructing an accurate and parsimonious symbolic expression of the governing equations. Physics-informed machine learning can combine the data-driven statistical inference capabilities of machine learning with traditional, knowledge-based science and engineering methods to improve the modeling of dynamical systems. In this presentation, we focus on using physics-informed machine learning to model Hamiltonian systems, which are a class of physical, classical, dynamical systems that conserve energy. We use Hamiltonian machine learning models, which are physics-informed machine learning models informed by Hamilton's equations. They are designed to learn the conserved quantity from which the systems' dynamics derive.

Our presentation first considers Hamiltonian neural networks. We propose informing Hamiltonian neural networks of additional physical properties of a system using observational, learning, and inductive biases. These properties include additive separability and periodicity. We then consider the case of raw observations of the Hamiltonian system taken in a non-canonical referential. This referential may have more dimensions than the system's degrees of freedom, or its basis may not directly comprise the canonical variables. We propose Hamiltonian machine learning models that leverage representation learning and symbolic regression, and comparatively and empirically evaluate them against state-of-the-art baselines. We show that the proposed models are more effective for the numerical regression and analytical interpretation of Hamiltonian systems.