Physics-Inspired Evolutionary Machine Learning Method: From the Schrödinger Equation to an Orbital-Free-DFT Kinetic...

Area of Science:

• Computational Physics and Chemistry
• Machine Learning in Scientific Discovery
• Quantum Mechanics and Electronic Structure Theory

Background:

• Deriving fundamental physical laws from observational data is a long-standing scientific challenge.
• Traditional methods often rely on human intuition and predefined theoretical frameworks.
• Machine learning offers new paradigms for scientific discovery, but integrating it with established physical principles remains an active research area.

Purpose of the Study:

• To introduce a novel machine learning (ML)-supervised evolutionary method, ML-Ω, inspired by physics' variational principles.
• To demonstrate ML-Ω's capability to derive underlying differential equations and functionals directly from data.
• To apply ML-Ω for developing advanced functionals in density functional theory (DFT).

Main Methods:

• Developed ML-Ω, a hypothesis evolutionary method that learns model functions (functionals) from data.
• Trained ML-Ω with minimal datasets: three hydrogen-like atom energies to rediscover Schrödinger's equation and functional.
• Trained ML-Ω with Thomas-Fermi (TF) energies to derive the exact TF functional.
• Applied ML-Ω to derive a local orbital-free (OF) kinetic energy functional (T_s) using energies of five atoms.

Main Results:

• ML-Ω successfully derived Schrödinger's exact functional and equation from limited atomic energy data.
• The method accurately reproduced the exact Thomas-Fermi functional.
• An ML-Ω-derived local OF-DFT functional (γTFλvW(0.964,1/4)) outperformed existing OF-DFT functionals.
• The new functional improved the description of stretched bonds in diatomic molecules compared to LDA and some GGA functionals.

Conclusions:

• The ML-Ω evolutionary method provides a powerful framework for discovering fundamental equations and functionals from data.
• This approach successfully bridges machine learning with natural sciences, enabling the derivation of complex physical laws.
• ML-Ω demonstrates significant potential for advancing electronic structure calculations and other scientific domains by generating accurate and efficient theoretical models.