Fig. 1: Workflow of the GC-(PI)HMC method and training workflow of the DP-Ne MLP.
a Workflow of the GC-(PI)HMC method. The trial moves for different types of degrees of freedom are selected based on a preset ratio using a random variable \(\xi\) satisfying a uniform distribution within \([0,\,1]\), and the probabilities of making trial moves for the internal degrees of freedom within the quantized beads’ configurations in PIMC \(({{\bf{R}}}_{\Delta }^{\left(k\right)})\), centroid atomic coordinates (R), total number of electrons (\({N}_{{\rm{e}}}\)) are \({a}_{1},{a}_{2}-{a}_{1},\,1-{a}_{2}\), respectively. b Construction framework and training workflow of the DP-Ne MLP adopted in this work. Initially, a data set is provided, followed by an iterative process which automatically goes through training, exploration, and labeling steps. The iteration is considered converged after the accurate sample percentage among the newly explored configurations is above 85%. The zoom-in schematic plot above the training workflow illustrates the construction framework of the DP-Ne force field. Atomic coordinates R of a modeling system are inputs for the embedding network generating descriptors \(\{{D}_{i}\}\). The fitting network maps \(\{{D}_{i}\}\) together with an extra degree of freedom \({N}_{{\rm{e}}}\) to the total energy E, atomic forces {\({\bf{F}}_{\bf{i}}\)}, and \(\partial E/\partial {N}_{{\rm{e}}}\) which relates to the work function of this extended configuration [R, \({N}_{{\rm{e}}}\)] as discussed in the following sections.
