Addressing Ill-conditioning in Density Functional Theory for Reliable Machine Learning

In principle, machine learning (ML) can be used to obtain any electronic property of a many-body system from its electron density within density functional theory. However, some physical quantities are highly sensitive to small variations in the density. This 'ill-conditioning' limits the accuracy with which these quantities can be learned as density functionals from a fixed amount of data. We identify sources of ill-conditioning present in density functionals that belong to two ubiquitous classes: 1) Physical quantities that are globally gauge-dependent, meaning they change value if a constant shift is applied to the external potential -- for example, the total energy; 2) Functionals of the N-electron density that have an implicit dependence on the (N+1)-electron density, such as the fundamental gap. We demonstrate that widely used ML models exhibit orders-of-magnitude greater error when applied to these ill-conditioned density functionals compared to other functionals that fall into neither class, even when the global gauge is fixed to prevent constant shifts. Owing to an absence of ill-conditioning in potential functionals, we find that providing the external potential as input to the ML model leads to significantly improved predictions of quantities in these two classes.