W$, evaluated at specially displaced atomic positions, $\mathbf r \mapsto \mathbf r \pm \epsilon \delta \mathbf f$, with $\epsilon$ a small parameter and $\delta \mathbf f = \hat{\mathbf f} - \mathbf f$ the observed force error. Details will be presented in a future work.[^1]

To improve the quality of our predictions, a single ANI model actually employs ensemble-averaging over 8 neural networks. Each neural network in the ensemble is trained to the same data, but using an independent random initialization of the model parameters. We observe that ensemble-averaged energy and force errors can be up to 20% and 40% smaller, respectively, than those of a single neural network prediction.

Active learning {#meth:activelearn}
---------------

![Diagram of the active learning sampling algorithm employed in this work. Multiple such cycles can be run simultaneously, with occasional synchronization points to combine all new data into a single global dataset. The MD sampling, DFT data generation, and ML model training all benefit from GPU-acceleration.[]{data-label="fig:aldatagen"}](images/AL-datageneration.png){width="1.0\columnwidth"}

### Overview

The active learning process employed here is similar to that in previous work, [@Smith2018] adapted for materials problems and efficient parallel execution on hundreds to thousands of nodes on the Sierra supercomputer. We first train an initial model to 