Best of both worlds: Enforcing detailed balance in machine learning models of transition rates

The slow microstructural evolution of materials often plays a key role in determining material properties. When the unit steps of the evolution process are slow, direct simulation approaches such as molecular dynamics become prohibitive and Kinetic Monte-Carlo (kMC) algorithms, where the state-to-state evolution of the system is represented in terms of a continuous-time Markov chain, are instead frequently relied upon to efficiently predict long-time evolution. The accuracy of kMC simulations however relies on the complete and accurate knowledge of reaction pathways and corresponding kinetics. This requirement becomes extremely stringent in complex systems such as concentrated alloys where the astronomical number of local atomic configurations makes the a priori tabulation of all possible transitions impractical. Machine learning models of transition kinetics have been used to mitigate this problem by enabling the efficient on-the-fly prediction of kinetic parameters. While conventional KMC methods based on transition state theory naturally yield reversible dynamics that exactly obey the detailed balance criterion, providing strong guarantees on the properties of the stationary distribution, many recently-proposed ML-based approaches to barrier predictions provide no such guarantees. In this study, we derive conditions under which physics-informed ML architectures exactly enforce the detailed balance condition by construction, even when relying on non-extensive descriptions of states in terms of local environments around mobile defects. Using the diffusion of a vacancy in a concentrated alloy as an example, we show that such ML architectures also exhibit superior performance in terms of prediction accuracy, demonstrating that the imposition of physical constraints can facilitate the accurate learning of barriers at no increase in computational cost.