Bypassing Backmapping: Coarse-Grained Electronic Property Distributions Using Heteroscedastic Gaussian Processes

We employ deep kernel learning electronic coarse-graining (DKL-ECG) with approximate Gaussian Processes as a flexible and scalable framework for learning heteroscedastic electronic property distributions as a smooth function of coarse-grained (CG) configuration. The appropriateness of the Gaussian prior on the predictive CG property distributions is justified as a function of CG model resolution by examining the statistics of the target distributions. The certainties of the predictive CG distributions are shown to be limited by CG model resolution, with DKL-ECG predictive noise converging to the intrinsic physical noise induced by the CG mapping operator for multiple chemistries. Further analysis of the resolution dependence of the learned CG property distributions allows for the identification of CG mapping operators that capture CG degrees of freedom with strong electron-phonon coupling. We further demonstrate the ability to construct the exact quantum chemical valence electronic density of states (EDOS), including behavior in the tails of the EDOS, from an entirely CG model by combining iterative boltzmann inversion and DKL-ECG. DKL-ECG provides a means of learning CG distributions of all-atom properties that are traditionally ``lost" in CG model development, introducing a robust methodological alternative to backmapping algorithms commonly employed to recover all-atom property distributions from CG simulations.