# Evolving symbolic density functionals

arXiv: https://arxiv.org/abs/2203.02540 · HTML: https://mlsanity.com/paper/2203.02540/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 85 references that are in the map.
- First posted on arXiv 3 Mar 2022.

## Builds on (most cited first, 13 of 13)

- [Regularized Evolution for Image Classifier Architecture Search](https://mlsanity.com/paper/1802.01548/) (2018)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [AutoML-Zero: Evolving Machine Learning Algorithms From Scratch](https://mlsanity.com/paper/2003.03384/) (2020)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [Evolving Reinforcement Learning Algorithms](https://mlsanity.com/paper/2101.03958/) (2021)
- [Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics](https://mlsanity.com/paper/2009.08551/) (2020)
- [Fast, accurate, and transferable many-body interatomic potentials by symbolic regression](https://mlsanity.com/paper/1904.01095/) (2019)
- [Learning the exchange-correlation functional from nature with fully differentiable density functional theory](https://mlsanity.com/paper/2102.04229/) (2021)
- [DeePKS: a comprehensive data-driven approach towards chemically accurate density functional theory](https://mlsanity.com/paper/2008.00167/) (2020)
- [Understanding Machine-learned Density Functionals](https://mlsanity.com/paper/1404.1333/) (2014)
- [Analytical classical density functionals from an equation learning network](https://mlsanity.com/paper/1910.12752/) (2019)
- [Neural-Guided Symbolic Regression with Asymptotic Constraints](https://mlsanity.com/paper/1901.07714/) (2019)

## Built on by (most cited first, 19 of 19)

- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [Transformer-based Planning for Symbolic Regression](https://mlsanity.com/paper/2303.06833/) (2023)
- [Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl](https://mlsanity.com/paper/2305.01582/) (2023)
- [Deep Generative Symbolic Regression with Monte-Carlo-Tree-Search](https://mlsanity.com/paper/2302.11223/) (2023)
- [Controllable Neural Symbolic Regression](https://mlsanity.com/paper/2304.10336/) (2023)
- [Opening the AI black box: program synthesis via mechanistic interpretability](https://mlsanity.com/paper/2402.05110/) (2024)
- [Image Super-resolution Inspired Electron Density Prediction](https://mlsanity.com/paper/2402.12335/) (2024)
- [(X)-evolve: Solution space evolution powered by large language models](https://mlsanity.com/paper/2508.07932/) (2025)
- [Evolutionary System Prompt Learning for Reinforcement Learning in LLMs](https://mlsanity.com/paper/2602.14697/) (2026)
- [A Transferable Recommender Approach for Selecting the Best Density Functional Approximations in Chemical Discovery](https://mlsanity.com/paper/2207.10747/) (2022)
- [Machine learning for accuracy in density functional approximations](https://mlsanity.com/paper/2311.00196/) (2023)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [EGG-SR: Embedding Symbolic Equivalence into Symbolic Regression via Equality Graph](https://mlsanity.com/paper/2511.05849/) (2025)
- [Weights to Code: Extracting Interpretable Algorithms from the Discrete Transformer](https://mlsanity.com/paper/2601.05770/) (2026)
- [Grad DFT: a software library for machine learning enhanced density functional theory](https://mlsanity.com/paper/2309.15127/) (2023)
- [Generalizability of Functional Forms for Interatomic Potential Models Discovered by Symbolic Regression](https://mlsanity.com/paper/2210.15124/) (2022)
- [OptAgent: Optimizing Query Rewriting for E-commerce via Multi-Agent Simulation](https://mlsanity.com/paper/2510.03771/) (2025)
- [Kohn-Sham Hamiltonian from Effective Field Theory: Quasiparticle Band Narrowing from Frozen Core Dynamics](https://mlsanity.com/paper/2604.25199/) (2026)
- [Agentic Discovery of Exchange-Correlation Density Functionals](https://mlsanity.com/paper/2605.05460/) (2026)

## Abstract (arXiv)

Systematic development of accurate density functionals has been a decades-long challenge for scientists. Despite the emerging application of machine learning (ML) in approximating functionals, the resulting ML functionals usually contain more than tens of thousands parameters, which makes a huge gap in the formulation with the conventional human-designed symbolic functionals. We propose a new framework, Symbolic Functional Evolutionary Search (SyFES), that automatically constructs accurate functionals in the symbolic form, which is more explainable to humans, cheaper to evaluate, and easier to integrate to existing density functional theory codes than other ML functionals. We first show that without prior knowledge, SyFES reconstructed a known functional from scratch. We then demonstrate that evolving from an existing functional ωB97M-V, SyFES found a new functional, GAS22 (Google Accelerated Science 22), that performs better for the majority of molecular types in the test set of Main Group Chemistry Database (MGCDB84). Our framework opens a new direction in leveraging computing power for the systematic development of symbolic density functionals.