Learning System Parameters from Turing Patterns

The Turing mechanism describes the emergence of spatial patterns due to spontaneous symmetry breaking in reaction-diffusion processes and underlies many developmental processes. Identifying Turing mechanisms in biological systems defines a challenging problem. This paper introduces an approach to the prediction of Turing parameter values from observed Turing patterns. The parameter values correspond to a parametrized system of reaction-diffusion equations that generate Turing patterns as steady state. The Gierer-Meinhardt model with four parameters is chosen as a case study. A novel invariant pattern representation based on resistance distance histograms is employed, along with Wasserstein kernels, in order to cope with the highly variable arrangement of local pattern structure that depends on the initial conditions which are assumed to be unknown. This enables to compute physically plausible distances between patterns, to compute clusters of patterns and, above all, model parameter prediction: for small training sets, classical state-of-the-art methods including operator-valued kernels outperform neural networks that are applied to raw pattern data, whereas for large training sets the latter are more accurate. Excellent predictions are obtained for single parameter values and reasonably accurate results for jointly predicting all parameter values.