Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations

TL;DR

This paper introduces hidden physics models that leverage Gaussian processes to efficiently learn nonlinear partial differential equations from small datasets, enabling data-driven discovery across scientific domains.

Contribution

The paper presents a novel Gaussian process-based framework for learning PDEs from limited data, integrating physics laws into machine learning models for improved data efficiency.

Findings

Successfully learned PDEs for Navier-Stokes, Schrödinger, Kuramoto-Sivashinsky, and fractional equations.

Demonstrated data-efficient PDE discovery in high-dimensional scientific problems.

Provided a new approach combining classical physics and modern machine learning.

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 {\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, Schr\"odinger, 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.