Efficient density functional theory learning with controllable predictive errors: What matters and what counts

Classical density functional theory (cDFT) is widely used in calculating the thermodynamic properties of interacting particles and inhomogeneous fluids. Machine learning strategies have been developed for learning several maps in cDFT calculation, including the maps from external and chemical potentials to particle density and potential energy, and from local density to one body correlation function. These new strategies hold great promises for physicists and chemists to overcome bottlenecks for simulating large-scale, complex systems that could otherwise be prohibitively expensive to compute. However, the factors and conditions that influence these machine learning approaches' performance are poorly understood. The goal of work is to explore the optimal strategy for learning functionals efficiently and reliably. First, how do we know what functionals of a particular system can be more efficiently learned? We demonstrated that the quantified uncertainty from the surrogate model can inform us of the most efficient strategy between several possible choices. Second, how do we reduce the computational cost from ML methods when the number of observations is large? By using the testbed of predicting one-body correlation functionals from local density profiles, we develop an efficient way based on a robust Gaussian process surrogate model that disentangles redundant information in neighboring density observations and thus substantially reduces the computational cost and enhances the robustness of the surrogate model. Finally, we developed active learning strategies that can theoretically control the predictive error of one-body correlation functionals from the external and chemical potentials or the local density. Our results are crucial for building reliable cyberinfrastructure for predicting functionals with a controlled predictive error.