Accelerating superconductor discovery through tempered deep learning of the electron-phonon spectral function

Integrating deep learning with the search for new electron-phonon superconductors represents a burgeoning field of research, where the primary challenge lies in the computational intensity of calculating the electron-phonon spectral function, $\alpha^2F(\omega)$, the essential ingredient of Midgal-Eliashberg theory of superconductivity. To overcome this challenge, we adopt a two-step approach. First, we compute $\alpha^2F(\omega)$ for 818 dynamically stable materials. We then train a deep-learning model to predict $\alpha^2F(\omega)$, using an unconventional training strategy to temper the model's overfitting, enhancing predictions. Specifically, we train a Bootstrapped Ensemble of Tempered Equivariant graph neural NETworks (BETE-NET), obtaining an MAE of 0.21, 45 K, and 43 K for the Eliashberg moments derived from $\alpha^2F(\omega)$: $\lambda$, $\omega_{\log}$, and $\omega_{2}$, respectively, yielding an MAE of 2.5 K for the critical temperature, $T_c$. Further, we incorporate domain knowledge of the site-projected phonon density of states to impose inductive bias into the model's node attributes and enhance predictions. This methodological innovation decreases the MAE to 0.18, 29 K, and 28 K, respectively, yielding an MAE of 2.1 K for $T_c$. We illustrate the practical application of our model in high-throughput screening for high-$T_c$ materials. The model demonstrates an average precision nearly five times higher than random screening, highlighting the potential of ML in accelerating superconductor discovery. BETE-NET accelerates the search for high-$T_c$ superconductors while setting a precedent for applying ML in materials discovery, particularly when data is limited.