We now assume we have reached a point in time t upd during the dynamics, where the TD-ADGA analysis finds that an update of the set of SPs are required. The question is then how to update the PES. Even though the change in the PES should primarily be significant in regions where the density has not yet grown large, simply replacing the old PES with the new PES at t upd can lead to non-physical behaviour as the propagation of the density will have been on account on the old PES. Furthermore, changing the PES will result in loss of the exact conservation of energy. To mitigate these challenges we propose two different schemes for updating the PES in a dynamics simulation.