Data-driven discovery of PDEs in complex datasets

TL;DR

This paper presents a deep learning approach to automatically discover PDEs from complex measurement data, including real-world weather data, by transforming and analyzing the data to reveal underlying physical laws.

Contribution

It introduces a novel deep learning method for PDE discovery that handles complex datasets and demonstrates its effectiveness on both model problems and real weather data.

Findings

Deep learning can accurately identify PDEs from measurement data.

Coordinate transformations are crucial for PDE discovery.

The method successfully applied to weather temperature distribution data.

Abstract

Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities of interest. A different approach is to measure the quantities of interest and use deep learning to reverse engineer the PDEs which are describing the physical process. In this paper we use machine learning, and deep learning in particular, to discover PDEs hidden in complex data sets from measurement data. We include examples of data from a known model problem, and real data from weather station measurements. We show how necessary transformations of the input data amounts to coordinate transformations in the discovered PDE, and we elaborate on feature and model selection. It is shown that the dynamics of a non-linear, second order PDE can be accurately described by an ordinary differential equation which is automatically discovered by our deep learning algorithm. Even more interestingly, we show that similar results apply in the context of more complex simulations of the Swedish temperature distribution.