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

We introduce a machine learning (ML)-supervised model function (which is in fact a functional rather than a regular function) that is inspired by the variational principle of physics. This ML hypothesis evolutionary method, termed ML-Ω, allows us to go from data to differential equation(s) underlying the physical (chemical, engineering, etc.) phenomena from which the data are derived from. The fundamental equations of physics can be derived from this ML-Ω evolutionary method when the proper training data is used. By training the ML-Ω model function with only three hydrogen-like atom energies, the method can find Schrödinger's exact functional and, from it, Schrödinger's fundamental equation. Then, in the field of density functional theory (DFT), when the model function is trained with the energies from the known Thomas-Fermi (TF) formula <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>E</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mn>0.7687</mml:mn><mml:msup><mml:mi>Z</mml:mi><mml:mrow><mml:mn>7</mml:mn><mml:mo>/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>, it correctly finds the exact TF functional. Finally, the method is applied to find a local orbital-free (OF) functional expression of the independent electron kinetic energy functional <i>T</i>_s based on the γ<i>TF</i>λ<i>vW</i> model. By considering the theoretical energies of only five atoms (He, Be, Ne, Mg, and Ar) as the training set, the evolutionary ML-Ω method finds an ML-Ω-OF-DFT local <i>T</i>_s functional (γ<i>TF</i>λ<i>vW</i>(0.964,1/4)) that outperforms all the OF-DFT functionals of a representative group. Moreover, our ML-Ω-OF functional overcomes the difficulty of LDA's and some local generalized gradient approximation (GGA)-DFT's functionals to describe the stretched bond region at the correct spin configuration of diatomic molecules. Nonsmooth and nonclosed form functionals can be considered in the ML-Ω model function and still be effectively trained. Although our evolutionary ML-Ω model function can work without an explicit prior-form functional, by using the techniques of symbolic regression, in this work, we exploit prior-form functional expressions to make the training process simpler and faster. The ML-Ω method can be considered at the intersection of ML and the natural sciences.