Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
This addresses the challenge of data scarcity in scientific domains by enabling data-efficient learning of PDEs, offering a new direction for applying classical methods to design learning machines without large datasets.
The authors tackled the problem of learning nonlinear partial differential equations from small, expensive data by introducing hidden physics models, which leverage underlying physical laws to extract patterns from high-dimensional experimental data, demonstrating effectiveness on canonical problems like Navier-Stokes and Schrödinger equations.
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 {\em small} data. In particular, we introduce \emph{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.