Advection-Diffusion on Graphs: A Bakry-Émery Laplacian for Spectral Graph Neural Networks

Graph Neural Networks (GNNs) often struggle to propagate information across long distances due to oversmoothing and oversquashing. Existing remedies such as graph transformers or rewiring typically incur high computational cost or require altering the graph structure. We introduce a Bakry-Émery graph Laplacian that integrates diffusion and advection through a learnable node-wise potential, inducing task-dependent propagation dynamics without modifying topology. This operator has a well-behaved spectral decomposition and acts as a drop-in replacement for standard Laplacians in spectral GNNs. Building on this insight, we develop µ-ChebNet, a spectral architecture that jointly learns the potential and Chebyshev filters, effectively bridging message-passing adaptivity and spectral efficiency. Our theoretical analysis shows how the potential modulates the spectrum, enabling control of key graph properties. Empirically, µ-ChebNet delivers consistent gains on synthetic long-range reasoning tasks, as well as real-world benchmarks, while offering an interpretable routing field that reveals how information flows through the graph. This establishes the Bakry-Émery Laplacian as a principled and efficient foundation for adaptive spectral graph learning.