Data-efficient prediction of rare earth leaching from industrial residues with physics-constrained machine learning

Shahram Asgari 

Coal ash, mine-water sludge, red mud and phosphogypsum contain rare earth elements, and each new feedstock needs its own leaching recipe. LixaLearn couples a small neural network to closed-form rate laws for four metal hosts (exchange sites, fast-dissolving minerals, slow-dissolving minerals and a locked fraction), with part of the fast host limited by the solubility of its mineral. The network maps bulk chemistry to 17 rate-law parameters, and one to three laboratory test runs on a new feedstock update them. On 175 synthetic feedstocks of seven types, the error on unseen recipes after three test runs was 5.4 percentage points, against 6.7 for tuned gradient boosting and 5.7 for a network without rate laws. The method was then developed on real data in the open. An earlier version failed a pre-registered test on 57 phosphogypsum runs, because its rate laws had no solubility limit; with a solubility-limited host and the common-ion effect of sulfate, three test runs predicted the other runs to 7.4 points, against 10.9 for gradient boosting and 11.5 for the network without rate laws (a development result). In the final held-out test, on 29 runs of an ion-adsorption clay, LixaLearn beat gradient boosting (3.3 against 5.5 points) and the direct network (12.3), but not the mean of the three test runs (2.7), and it did not rank the clay runs. On red mud leached with citric acid, the machine-learning models remained better. Physics-constrained learning transferred to real feedstocks where its rate laws had the right structure, and checking that structure on a few complete runs should come before using it on a new kind of feedstock.