Geochemical modeling of the mantle partial melting using heuristic exploration: an optimization model applied to earth sciences

An evolution strategy-type heuristic simulation tool was developed to model the inverse problem of partial melting. The complexity of such a problem is expressed by the non-uniqueness of solution and the high number of variables it could have: 1 denoting the degree of freedom + \(\varvec{m}\) from the number of minerals of the rock + \(\varvec{q}\) from the number of rare earth elements. This motivated us to redefine the problem as an optimization one deriving an objective function from the batch equation to model the source composition and the partial melting degree from the chemistry of near-primary liquids. Then, we optimize the inverse modeling by using a heuristic approach, namely, Evolution Strategy. In the search algorithm structure it was considered: (a) the geochemical system constraints, (b) an initialization step, and (c) a procedure of mutation and heuristic individual selection. The heuristic simulation was successfully applied in four study cases, as mineralogical and rare earth element (REE) composition of known peridotitic sources. The partial melting conditions were reproduced with a deviation \(\varvec{\le } \varvec{10^{-6}}\) in a reasonably practical time (\(\varvec{\sim 2}\) hours), in the search for 1000 solutions, by using a desktop computer. Moreover, our technique provides several efficient solutions, due to non-uniqueness, which allows to explore different scenarios when when a priori knowledge is missing. On the other hand, the algorithm is also able to approach the original melt when knowing only \(\varvec{n+1}\) variables of it. The flexibility in the construction of our complete methodology, joined to the successfully results and the possibility to apply it to any trace element, suggests its extension to similar magmatic processes.