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.

整合深度学习与寻找新的电声子超导体的研究代表了一个蓬勃发展的领域，其中主要挑战在于计算电声子谱函数α²F(ω)，超导理论Midgal-Eliashberg的主要组成部分。为了克服这一挑战，我们采用两步法，首先计算818种动态稳定材料的α²F(ω)，然后使用一种非常规的训练策略训练一个深度学习模型BETE-NET来预测α²F(ω)，并通过增加模型的归一化改进预测。此外，我们还将声子密度的领域知识结合到模型的节点属性中，提高预测精度。该方法的创新降低了临界温度$T_c$的平均绝对误差（MAE）至2.1K，对高温超导体材料的高通量筛选展示了实际应用。BETE-NET在加速高温超导体的搜索同时为材料发现中应用机器学习树立了先例，特别是在数据有限的情况下。

通过温和的电子-声子谱函数深度学习加速超导体发现