Physics-Inspired\nEvolutionary Machine Learning Method:\nFrom the Schrodinger Equation to an Orbital-Free-DFT Kinetic\nEnergy Functional

We introduce a machine learning (ML)-supervised model\nfunction\n(which is in fact a functional rather than a regular function) that\nis inspired by the variational principle of physics. This ML hypothesis\nevolutionary method, termed ML-Ω, allows us to go from data\nto differential equation(s) underlying the physical (chemical, engineering,\netc.) phenomena from which the data are derived from. The fundamental\nequations of physics can be derived from this ML-Ω evolutionary\nmethod when the proper training data is used. By training the ML-Ω\nmodel function with only three hydrogen-like atom energies, the method\ncan find Schrödinger’s exact functional and, from it,\nSchrödinger’s fundamental equation. Then, in the field\nof density functional theory (DFT), when the model function is trained\nwith the energies from the known Thomas−Fermi (TF) formula E=−0.7687Z7/3, it correctly finds the exact TF functional.\nFinally, the method is applied to find a local orbital-free (OF) functional\nexpression of the independent electron kinetic energy functional <i>T</i><sub>s</sub> based on the γ<i>TF</i>λ<i>vW</i> model. By considering the theoretical energies of only\nfive atoms (He, Be, Ne, Mg, and Ar) as the training set, the evolutionary\nML-Ω method finds an ML-Ω-OF-DFT local <i>T</i><sub>s</sub> functional (γ<i>TF</i>λ<i>vW</i>(0.964,1/4)) that outperforms all the OF-DFT functionals\nof a representative group. Moreover, our ML-Ω-OF functional\novercomes the difficulty of LDA’s and some local generalized\ngradient approximation (GGA)-DFT’s functionals to describe\nthe stretched bond region at the correct spin configuration of diatomic\nmolecules. Nonsmooth and nonclosed form functionals can be considered\nin the ML-Ω model function and still be effectively trained.\nAlthough our evolutionary ML-Ω model function can work without\nan explicit prior-form functional, by using the techniques of symbolic\nregression, in this work, we exploit prior-form functional expressions\nto make the training process simpler and faster. The ML-Ω method\ncan be considered at the intersection of ML and the natural sciences.