The Bigger the Better? Accurate Molecular Potential Energy Surfaces from Minimalist Neural Networks

TL;DR

KerNN, a combined kernel/neural network-based approach to represent molecular PESs, is introduced and shows excellent performance on test set statistics and observables including vibrational bands computed from classical and quantum simulations.

Abstract

Atomistic simulations are a powerful tool for studying the dynamics of molecules, proteins, and materials on wide time and length scales. Their reliability and predictiveness, however, depend directly on the accuracy of the underlying potential energy surface (PES). Guided by the principle of parsimony this work introduces KerNN, a combined kernel/neural network-based approach to represent molecular PESs. Compared to state-of-the-art neural network PESs the number of learnable parameters of KerNN is significantly reduced. This speeds up training and evaluation times by several orders of magnitude while retaining high prediction accuracy. Importantly, using kernels as the features also improves the extrapolation capabilities of KerNN far beyond the coverage provided by the training data which solves a general problem of NN-based PESs. KerNN applied to spectroscopy and reaction dynamics shows excellent performance on test set statistics and observables including vibrational bands computed from classical and quantum simulations.

Figures (15)

• Figure 1: Schematic representation of A) formaldehyde, B) the two reaction channels of the HeH$_2^+$ system and C) hydrogen oxalate (or deprotonated oxalic acid) in its cyclic/hydrogen bonded form.
• Figure 2: Energy and force learning curves for the different variants of the H$_2$CO PESs trained on CCSD(T)-F12B/aug-cc-pVTZ-F12 reference data. These are compared to PhysNet results taken from Reference MM.h2co:2020. Solid and dashed lines represent MAEs and RMSEs, respectively. A total of five KerNN models were trained for each value of $N_{\rm Train}$ on different splits of the data and only the mean out-of-sample errors are shown. KerNN$^{\rm ns}$ and KerNN$^{\rm s}$ represent the NNs that use the non-symmetrized ($\mathcal{D}^{\rm ns}$) and symmetrized ($\mathcal{D}^{\rm s}$) descriptors, respectively. The flattening in energies for $N_{\rm Train} \geq 1600$ is caused by the "error floor" noted in earlier work for the CCSD(T)-F12 forces.MM.h2co:2020 Note that the different models can be more or less sensitive to such noise and therefore exhibit a flattening at higher/lower test set errors. The lowest test set MAEs reported in Reference MM.h2co:2020, for example, were MAE($E$) = 3E-4 and MAE($F$)=1E-4.
• Figure 3: Atomic MAE($\bm{F}$) (i.e. the mean absolute error between the reference and predicted forces for each of the atoms) for the symmetric structure with short H-C bonds shown. The corresponding (aggregated) MAE($\bm{F}$) are 1.3/0.3 kcal/mol/Å. While KerNN$^{\rm s}$ predicts symmetrical errors, this is not so for KerNN$^{\rm ns}$.
• Figure 4: A: The extrapolation capabilities of the ML-PES are assessed on a data set containing 2500 structures, generated from normal mode sampling at a higher temperature (5000 K) than the training set ($2000$ K). The extrapolation data set was available from previous work.MM.h2co:2020 While the training data covers an energy range of roughly 40 kcal/mol the extrapolation data set covers 130 kcal/mol. B: One-dimensional PES cut along the C-H bond length for different ML models ($r$ and $e^{-r}$ correspond to NNs with the same architecture as KerNN, but employing different descriptors, namely the interatomic distances $r$ and $e^{-r}$). KerNN$^{\rm s}_{\rm TL}$ corresponds to a model with asymptotic behaviour adjusted according to the experimentally determined dissociation energy. The training data range is shaded in gray.
• Figure 5: Infrared spectra derived from finite-$T$ MD simulations of H$_2$CO. The experimental fundamentalsherndon:2005 are the grey Gaussians. The computed spectra were averaged over 100 independent trajectories, each 200 ps in length, using $\Delta t = 0.2$ fs.