Incorporating Detailed Balance and Uncertainty Quantification in Machine Learning-Based Models of Defect Transport in Alloys

With the increasing interest in so-called high entropy, multi-principal component, or compositionally complex alloys, there is a greater need to understand how transport is affected by the complex chemistry of these materials. As compared to a simple elemental solid, in which every lattice site is identical and transport can be described by a handful of saddle points, these materials exhibit a very rugged potential energy surface in which every site has a unique chemical environment. This means that every site exhibits a different defect formation energy and corresponding migration barriers for motion to neighboring sites. As the number of elements in these systems increases, it becomes increasingly challenging, and soon impossible, to enumerate the energetics of every site in the system. Alternative approaches are necessary.<br/> <br/>Machine learning has become the choice <i>du jour</i> to describe the chemistry-dependent properties of defects in these alloys. Using atomistic data as their foundation, machine learning models can then quickly and accurately predict the migration barriers for defects as a function of the local chemical environment, eliminating the need to determine all possible barriers a priori. As the basis of kinetic Monte Carlo simulations, these models then allow for long-time simulations of transport through the alloy. However, for such simulations to inspire confidence, they must obey detailed balance. Further, it would be beneficial to have some metric of the uncertainty in the predicted barriers and an approach to systematically improve them.<br/> <br/>Here, we describe approaches to address both questions. First, we describe how detailed balance can be rigorously implemented in machine learning models of defect energetics. Specifically, we theoretically derive the conditions under which detailed balance will be strictly satisfied and demonstrate that machine learning models based on this derivation do indeed obey detailed balance. We further show that less rigorous treatments lead to systematic errors that can lead to energy drift as the simulation proceeds. We conclude that detailed balance must be considered to obtain valid trajectories and that implementation of detailed balance is straightforward.<br/> <br/>We then use the machine learning model to determine the kinetic properties of defects. Using a recently-developed approach in which the complex correlated+uncorrelated kinetic Monte Carlo problem is mapped rigorously onto an uncorrelated surrogate, we then use this to identify which atomic scale events are most critical in describing the trajectory of the defect. This provides a route to close the loop, so to speak: to build an autonomous workflow in which the events that the trajectory is most sensitive to are further refined to improve the description of the material. We demonstrate the elements of this workflow, highlighting how we can then quickly determine the diffusion tensor for defects in complex alloys when given appropriate computational resources.