Reliable emulation of complex functionals by active learning with error control

Statistical emulator is a surrogate model of complex physical models to drastically reduce the computational cost. Its successful implementation hinges on the accurate representation of the nonlinear response surface with a high-dimensional input space. Conventional "space-filling" designs, including random sampling and Latin hypercube sampling, become inefficient as the dimensionality of the input variables increases and are problematic in the functional space. To address this fundamental challenge, we develop a reliable emulator for predicting complex functionals by active-learning with error control (ALEC) that is applicable to infinite-dimensional mapping with high-fidelity predictions and a controlled predictive error. The computational efficiency has been demonstrated by emulating the classical density functional theory (cDFT) calculations, a statistical-mechanical method widely used in modeling the equilibrium properties of complex molecular systems. We show that the ALEC emulator is much more accurate than conventional Gaussian processes emulators based on "space-filling" designs, another widely used active learning approach, and computationally more efficient than direct cDFT calculations. The ALEC framework can be a reliable building block for emulating expensive functionals, because of its reduced computational cost, controlled predictive error, and fully automatic features.