# Quantum deep field: data-driven wave function, electron density generation, and atomization energy prediction and extrapolation with machine learning

arXiv: https://arxiv.org/abs/2011.07923 · HTML: https://mlsanity.com/paper/2011.07923/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Nov 2020.

## Built on by (most cited first, 11 of 11)

- [Ab-Initio Potential Energy Surfaces by Pairing GNNs with Neural Wave Functions](https://mlsanity.com/paper/2110.05064/) (2021)
- [Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids](https://mlsanity.com/paper/2112.00652/) (2021)
- [Machine learning nonequilibrium electron forces for adiabatic spin dynamics](https://mlsanity.com/paper/2112.12124/) (2021)
- [Descriptors for Machine Learning Model of Generalized Force Field in Condensed Matter Systems](https://mlsanity.com/paper/2201.00798/) (2022)
- [Equivariant Neural Operator Learning with Graphon Convolution](https://mlsanity.com/paper/2311.10908/) (2023)
- [Audacity of huge: overcoming challenges of data scarcity and data quality for machine learning in computational materials discovery](https://mlsanity.com/paper/2111.01905/) (2021)
- [A Tutorial on the Use of Physics-Informed Neural Networks to Compute the Spectrum of Quantum Systems](https://mlsanity.com/paper/2407.20669/) (2024)
- [Machine Learning Time Propagators for Time-Dependent Density Functional Theory Simulations](https://mlsanity.com/paper/2508.16554/) (2025)
- [Machine learning nonequilibrium phase transitions in charge-density wave insulators](https://mlsanity.com/paper/2601.07583/) (2026)
- [Physics-Informed Neural Networks with Architectural Physics Embedding for Large-Scale Wave Field Reconstruction](https://mlsanity.com/paper/2603.02231/) (2026)
- [A cyclical route linking fundamental mechanism and AI algorithm: An example from tuning Poisson's ratio in amorphous networks](https://mlsanity.com/paper/2312.03404/) (2023)

## Abstract (arXiv)

Deep neural networks (DNNs) have been used to successfully predict molecular properties calculated based on the Kohn--Sham density functional theory (KS-DFT). Although this prediction is fast and accurate, we believe that a DNN model for KS-DFT must not only predict the properties but also provide the electron density of a molecule. This letter presents the quantum deep field (QDF), which provides the electron density with an unsupervised but end-to-end physics-informed modeling by learning the atomization energy on a large-scale dataset. QDF performed well at atomization energy prediction, generated valid electron density, and demonstrated extrapolation. Our QDF implementation is available at https://github.com/masashitsubaki/QuantumDeepField_molecule.