Learning partial differential equations from data

Abstract

Partial differential equations (PDEs) are ubiquitous in science and engineering for their ability to model the behavior of various systems. In science, PDEs are used to model a multitude of phenomena ranging from quantum mechanics to brain modeling. In engineering, PDEs form the basis of most simulation software which is used to model processes such as heat transfer and collapse of structures.

Many systems of interest already have accurate PDE-based models, but some systems are so complex that describing them in terms of PDEs possesses a serious challenge. This process might be simplified with the help of machine learning. When there is enough observations about a system, PDEs governing this system might be "learned".

This work proposes a method of learning black-box approximations of PDEs from data. The method is based on graph neural networks which allows it to be used on unstructured spatial grids. Furthermore, the continuous-time nature of the method makes it robust against perturbations in the time grid.

Experiments demonstrate that the method can be applied to different types of PDEs, can be used on solution domains of different shapes, supports different boundary conditions and is able to work with noisy data.