Addressing hypersensitivity in density functional theory for reliable machine learning

In principle, machine learning (ML) can be used within density functional theory to obtain any electronic property of a time-independent many-body system from its ground-state electron density alone. However, when the target quantity (the output) is highly sensitive to small variations in the density (the input), accurately learning the mapping between them is highly challenging in practice. We identify this 'hypersensitivity' in two classes of density functionals that contain observables central to most electronic structure calculations: (1) any observable that varies subject to a global constant shift in the external potential (such as the total energy); (2) any observable of the N-electron system that depends on the properties of the (N + 1)-electron system (such as the electron affinity). We demonstrate that the accuracy with which widely used ML models approximate density functionals in these two classes is significantly limited by the pathological nature of the mapping from the density to the observable, even when the reference level of the external potential is fixed to prevent constant shifts. In addition, we find that providing the external potential as input to the ML model resolves this issue, in some cases reducing the error by up to three orders of magnitude when predicting quantities in these two classes.