Datadriven discovery of PDEs in complex datasets
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 nonlinear, 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.
 Publication:

Journal of Computational Physics
 Pub Date:
 May 2019
 DOI:
 10.1016/j.jcp.2019.01.036
 arXiv:
 arXiv:1808.10788
 Bibcode:
 2019JCoPh.384..239B
 Keywords:

 Machine learning;
 Deep learning;
 Partial differential equations;
 Neural networks;
 Statistics  Machine Learning;
 Computer Science  Machine Learning;
 Mathematics  Numerical Analysis
 EPrint:
 doi:10.1016/j.jcp.2019.01.036