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that the traditional volume or ensemble averaging of the equations of continuum mechanics over the heterogeneity is no longer valid and, therefore, the issue of discovering the governing equations for flow and transport processes is an open question.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.

Figure 1

Comparison of the predictions of the discovered fractional diffusion equation (curves) with the results of numerical simulations of diffusion on the critical percolation cluster at the percolation threshold: (a) Eq. (5) and (b) Eq. (7). To discover the equation, the last 40% of the data at the longest times were used; circle. Triangles and squares represent the numerical results, while red, green, and blue show the model's predictions.

Figure 2

Comparison of the predictions of the discovered fractional diffusion equation (curves) with the results of numerical simulations of diffusion on the critical percolation cluster at the percolation threshold over the entire simulation time: (a) Eq. (5) and (b) Eq. (7). Diamond, cross, circle, triangle, and square represent the numerical results while cyan, magenta, red, green, and blue show the predictions.