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Networks are abundant in the life sciences. Outstanding challenges include how to characterize similarities between networks, and in extension how to integrate information across networks. Yet, network alignment remains a core algorithmic problem. Here, we present a novel learning algorithm called evolutionary heat diffusion-based network alignment (EDNA) to address this challenge. EDNA uses the diffusion signal as a proxy for computing node similarities between networks. Comparing EDNA with state-of-the-art algorithms on a popular protein-protein interaction network dataset, using four different evaluation metrics, we achieve (i) the most accurate alignments, (ii) increased robustness against noise, and (iii) superior scaling capacity. The EDNA algorithm is versatile in that other available network alignments/embeddings can be used as an initial baseline alignment, and then EDNA works as a wrapper around them by running the evolutionary diffusion on top of them. In conclusion, EDNA outperforms state-of-the-art methods for network alignment, thus setting the stage for large-scale comparison and integration of networks.