Data-driven discovery of the governing equations for transport in heterogeneous media by symbolic regression and stochastic optimization.

With advances in instrumentation and the tremendous increase in computational power, vast amounts of data are becoming available for many complex phenomena in macroscopically heterogeneous media, particularly those that involve flow and transport processes, which are problems of fundamental interest that occur in a wide variety of physical systems. The absence of a length scale beyond which such systems can be considered as homogeneous implies that the traditional volume or ensemble averaging of the equations of continuum mechanics over the heterogeneity is no longer valid and, therefore, the issue of discovering the governing equations for flow and transport processes is an open question. We propose a data-driven approach that uses stochastic optimization and symbolic regression to discover the governing equations for flow and transport processes in heterogeneous media. The data could be experimental or obtained by microscopic simulation. As an example, we discover the governing equation for anomalous diffusion on the critical percolation cluster at the percolation threshold, which is in the form of a fractional partial differential equation, and agrees with what has been proposed previously.