where F ω,i (x) is the force of atom i calculated by DP E ω . A minimization in ϵ and the prediction errors in the phonon density of states g(ω) resulted in a DP that was used to calculate the D s in DSG. If these tests failed, then configurations that were tagged as candidate configurations from the MD exploration step were relabeled with DFT energies and forces and added back to the database, thus leading to another active learning iteration. Our final DP for DSG was obtained after two active learning iterations, and was then used to compute all of our results for DSG. Center of Excess Charge. We developed a method to locate the position of a proton in DSG by tracking the center of excess charge (CEC). The CEC method for graphanol was first developed by Achar et al. to track protons in SSG . This same method was used to track protons in DSG. There have been other CEC tracking methods like Li and Swanson . These methods are more involved, as they explore multiple proton migration topologies to determine the CEC for a given configuration. In contrast, the CEC tracking method proposed here offers a straightforward option for graphanol by relying solely on geometric information. We defined the CEC in DSG as the position of the O atom that contains a proton, which is where the excess charge resides. Tracking oxygen atoms instead of the H atom is more convenient. This is because the identity of the proton changes from one H atom to another very rapidly, making explicit proton tracking problematic and subject to relatively large jumps. A molecular visualization of a state with a CEC in DSG is shown in Figure . Only the green O atom (labeled as A) is the CEC for that state. An O atom can be considered a CEC if it has two H atoms within a cutoff radius of 1.2 Å, indicated by the green dotted circle. There can be a situation where two or more O atoms have two H atoms within their respective cutoffs, such as the blue O atom in Figure d O-H/H + becomes the CEC for that instant. In the situation of Figure , A becomes the CEC, and B does not. An additional constraint was applied, where the CEC for the next time step (t + δt) can only be within the first nearest O neighbors of the CEC at the current time t. The CEC from t can be the CEC at t + δt. In the case of the state shown in Figure , the candidate CECs for the next time step are within the dotted red circle, which includes O atoms labeled A, B, C, D, E, F, and G. This constraint was implemented because within an MD time step of 0.25 fs, protons cannot hop across more than one O atom. Fig. : Molecular visualization of a state in DSG with an added proton. The CEC (labeled A) for that state (time t) is depicted as a green atom. A candidate CEC (labeled B) is colored blue. Corresponding green and blue dashed circles indicate the cutoff radius using to find nearest H neighbors. A dashed red circle centered around the CEC is a cutoff for the next CEC at time t + δt. Image was rendered using Ovito. for Research Computing (RRID:SCR 022735) H2P cluster, which is supported by NSF award number OAC-2117681.