Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations

TL;DR

The paper tackles learning time-dependent nonlinear PDEs from limited data by embedding physical laws into a Gaussian-process-based hidden physics model. It leverages backward Euler discretization to relate two close snapshots and learns PDE operator parameters $\lambda$ by minimizing the negative log marginal likelihood, yielding a flexible, data-efficient framework for PDE discovery and system identification. The method is demonstrated across Burgers', KdV, Kuramoto–Sivashinsky, nonlinear Schrödinger, Navier–Stokes, and fractional equations, showing robust parameter inference from very small datasets and explicit uncertainty quantification. This physics-informed approach offers a scalable, non-differentiation-based alternative for learning complex dynamical systems in data-scarce settings and provides public code resources for broader adoption.

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, 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.

Figures (7)

• Figure 1: Burgers' equation: A solution to the Burgers' equation is depicted in the top panel. The two white vertical lines in this panel specify the locations of the two randomly selected snapshots. These two snapshots are $\Delta t = 0.1$ apart and are plotted in the middle panel. The red crosses denote the locations of the training data points. The correct partial differential equation along with the identified ones are reported in the lower panel.
• Figure 2: The KdV equation: A solution to the KdV equation is depicted in the top panel. The two white vertical lines in this panel specify the locations of the two randomly selected snapshots. These two snapshots are $\Delta t = 0.1$ apart and are plotted in the middle panel. The red crosses denote the locations of the training data points. The correct partial differential equation along with the identified ones are reported in the lower panel.
• Figure 3: Kuramoto-Sivashinsky equation: A solution to the Kuramoto-Sivashinsky equation is depicted in the top panel. The two white vertical lines in this panel specify the locations of the two randomly selected snapshots. These two snapshots are $\Delta t = 0.4$ apart and are plotted in the middle panel. The red crosses denote the locations of the training data points. The correct partial differential equation along with the identified ones are reported in the lower panel.
• Figure 4: Nonlinear Schrödinger equation: A solution to the nonlinear Schrödinger equation is depicted in the top two panels. The two black vertical lines in these two panels specify the locations of the two randomly selected snapshots. These two snapshots are $\Delta t = 0.0063$ apart and are plotted in the two middle panels. The red crosses denote the locations of the training data points. The correct partial differential equation along with the identified ones are reported in the lower panel. Here, $u$ is the real part of $h$ and $v$ is the imaginary part.
• Figure 5: Navier-Stokes equations: A single snapshot of the vorticity field of a solution to the Navier-Stokes equations for the fluid flow past a cylinder is depicted in the top panel. The black box in this panel specifies the sampling region. Two snapshots of the velocity field being $\Delta t = 0.02$ apart are plotted in the two middle panels. The black crosses denote the locations of the training data points. The correct partial differential equation along with the identified ones are reported in the lower panel. Here, $u$ denotes the $x$-component of the velocity field, $v$ the $y$-component, $p$ the pressure, and $w$ the vorticity field.