Neural Integral Functionals

Zheyuan Hu, Tianbo Li, Zekun Shi, Kunhao Zheng, Giovanni Vignale, Kenji Kawaguchi, Shuicheng YAN, Min Lin

Keywords: Deep Learning, AI for Science, Functional, Density Functional Theory

TL;DR: We propose the neural integral functionals to approximate functionals and functional derivatives and to solve functional minimization problems.

Abstract: Functionals map input functions to output scalars, which are ubiquitous in various scientific fields. In this work, we propose neural integral functional (NIF), which is a general functional approximator that suits a large number of scientific problems including the brachistochrone curve problem in classical physics and density functional theory in quantum physics. One key ingredient that enables NIF on these problems is the functional’s explicit dependence on the derivative of the input function. We demonstrate that this is crucial for NIF to outperform neural operators (NOs) despite the fact that NOs are theoretically universal. With NIF, we further propose to jointly train the functional and its functional derivation (FD) to improve generalization and to enable applications that require accurate FD. We validate these claims with experiments on functional fitting and functional minimization.