Reliable emulation of complex functionals by active learning with error control

TL;DR

This paper introduces ALEC, an active learning-based emulator with error control, significantly improving accuracy and efficiency in modeling complex, high-dimensional functionals like classical density functional theory.

Contribution

The paper presents ALEC, a novel active learning algorithm that reliably emulates complex functionals with controlled error, outperforming traditional methods in accuracy and computational efficiency.

Findings

ALEC achieves higher accuracy than Gaussian process-based emulators.

ALEC is more computationally efficient than direct cDFT calculations.

ALEC effectively handles infinite-dimensional mappings with error control.

Abstract

A statistical emulator can be used as a surrogate of complex physics-based calculations to drastically reduce the computational cost. Its successful implementation hinges on an 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 the predictive accuracy of the emulator can degrade substantially for a test input distant from the training input set. To address this fundamental challenge, we develop a reliable emulator for predicting complex functionals by active learning with error control (ALEC). The algorithm 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 ALEC is much more accurate than conventional emulators based on the Gaussian processes with "space-filling" designs and alternative active learning methods. Besides, it is computationally more efficient than direct cDFT calculations. ALEC can be a reliable building block for emulating expensive functionals owing to its minimal computational cost, controllable predictive error, and fully automatic features.