On the Effect of Aleatoric and Epistemic Errors on the Learnability and Quality of NN-based Potential Energy Surfaces

The effect of noise in the input data for learning potential energy surfaces (PESs) based on neural networks for chemical applications is assessed. Noise in energies and forces can result from aleatoric and epistemic errors in the quantum chemical reference calculations. Statistical (aleatoric) noise arises for example due to the need to set convergence thresholds in the self consistent field (SCF) iterations whereas systematic (epistemic) noise is due to, {\it inter alia}, particular choices of basis sets in the calculations. The two molecules considered here as proxies are H$_{2}$CO and HONO which are examples for single- and multi-reference problems, respectively, for geometries around the minimum energy structure. For H$_2$CO it is found that adding noise to energies with magnitudes representative of single-point calculations does not deteriorate the quality of the final PESs whereas increasing the noise level commensurate with electronic structure calculations for more complicated, e.g. metal-containing, systems is expected to have a more notable effect. However, the effect of noise on the forces is more noticeable. On the other hand, for HONO which requires a multi-reference treatment, a clear correlation between model quality and the degree of multi-reference character as measured by the $T_1$ amplitude is found. It is concluded that for chemically "simple" cases the effect of aleatoric and epistemic noise is manageable without evident deterioration of the trained model - although the quality of the forces is important. However, considerably more care needs to be exercised for situations in which multi-reference effects are present.