Data-driven discovery of PDEs in complex datasets
This addresses the challenge of deriving PDEs from data for scientists and engineers, offering a data-driven alternative to traditional first-principles methods, though it appears incremental as it builds on existing deep learning approaches.
The paper tackles the problem of discovering partial differential equations (PDEs) from complex datasets, such as weather station measurements, using deep learning, and shows that it can accurately describe dynamics, including reducing a non-linear second-order PDE to an ordinary differential equation.
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.