Abstract
While there is currently a lot of enthusiasm about "big data", useful data is
usually "small" and expensive to acquire. In this paper, we present a new
paradigm of learning partial differential equations from small data. In
particular, we introduce 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 methodology may be applied to the problem of
learning, system identification, or data-driven discovery of partial
differential equations. Our framework relies on Gaussian processes, a powerful
tool for probabilistic inference over functions, that enables us to strike a
balance between model complexity and data fitting. The effectiveness of the
proposed approach is demonstrated through a variety of canonical problems,
spanning a number of scientific domains, including the Navier-Stokes,
Schrödinger, Kuramoto-Sivashinsky, and time dependent linear fractional
equations. The methodology provides a promising new direction for harnessing
the long-standing developments of classical methods in applied mathematics and
mathematical physics to design learning machines with the ability to operate in
complex domains without requiring large quantities of data.
Description
Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
Links and resources
Tags
community