Machine learning of superconducting critical temperature from Eliashberg theory

The Eliashberg theory of superconductivity accounts for the fundamental physics of conventional electron-phonon superconductors, including the retardation of the interaction and the effect of the Coulomb pseudopotential, to predict the critical temperature $T_c$ and other properties. McMillan, Allen, and Dynes derived approximate closed-form expressions for the critical temperature predicted by this theory, which depends essentially on the electron-phonon spectral function $\alpha^2F(\omega)$, using $\alpha^2F$ for low-$T_c$ superconductors. Here we show that modern machine learning techniques can substantially improve these formulae, accounting for more general shapes of the $\alpha^2F$ function. Using symbolic regression and the sure independence screening and sparsifying operator (SISSO) framework, together with a database of artificially generated $\alpha^2F$ functions, ranging from multimodal Einstein-like models to calculated spectra of polyhydrides, as well as numerical solutions of the Eliashberg equations, we derive a formula for $T_c$ that performs as well as Allen-Dynes for low-$T_c$ superconductors, and substantially better for higher-$T_c$ ones. The expression identified through our data-driven approach corrects the systematic underestimation of $T_c$ while reproducing the physical constraints originally outlined by Allen and Dynes. This equation should replace the Allen-Dynes formula for the prediction of higher-temperature superconductors and for the estimation of $\lambda$ from experimental data.