Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations

We introduce the concept of hidden physics models, which are essentially data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and nonlinear partial differential equations, to extract patterns from high-dimensional data generated from experiments. The proposed technology is applied to the problem of learning, system identification, or data-driven discovery of partial differential equations. The framework relies on Gaussian processes, a powerful tool for probabilistic inference over functions, to strike a balance between model complexity and data fit. The effectiveness of the proposed approach is demonstrated through a variety of canonical problems, spanning a number of scientific domains, including the Navier-Stokes, Schr\"{o}dinger, Kuramoto-Sivashinsky, and fractional equations. The methodology provides a promising new direction for capitalizing on the long-standing developments of classical methods in applied mathematics and mathematical physics to design learning machines with the ability to learn in complex domains without requiring large quantities of data.