Hidden physics models: Machine learning of nonlinear partial differential equations

Abstract

While there is currently a lot of enthusiasm about "big data", useful data is\nusually "small" and expensive to acquire. In this paper, we present a new\nparadigm of learning partial differential equations from {\\em small} data. In\nparticular, we introduce \\emph{hidden physics models}, which are essentially\ndata-efficient learning machines capable of leveraging the underlying laws of\nphysics, expressed by time dependent and nonlinear partial differential\nequations, to extract patterns from high-dimensional data generated from\nexperiments. The proposed methodology may be applied to the problem of\nlearning, system identification, or data-driven discovery of partial\ndifferential equations. Our framework relies on Gaussian processes, a powerful\ntool for probabilistic inference over functions, that enables us to strike a\nbalance between model complexity and data fitting. The effectiveness of the\nproposed approach is demonstrated through a variety of canonical problems,\nspanning a number of scientific domains, including the Navier-Stokes,\nSchr\\"odinger, Kuramoto-Sivashinsky, and time dependent linear fractional\nequations. The methodology provides a promising new direction for harnessing\nthe long-standing developments of classical methods in applied mathematics and\nmathematical physics to design learning machines with the ability to operate in\ncomplex domains without requiring large quantities of data.\n