METRO: Metric-Enhanced Token Routing Operator

Nodens Koren, Thomas Hofmann, Georgios Kissas

State-of-the-art neural operators scale to complex meshes via slice-and-process architectures, yet many rely on linear compatibility scores for latent tokenization. Under common feature normalization, such scores are equivalent to isotropic Euclidean clustering, while without normalization they induce unbounded linear decision regions. In both cases, they lack slice-specific anisotropic locality, which can lead to redundant and entangled latent slices. To address this, we propose Metric-Enhanced Token Routing Operator (METRO), a geometry-aware routing mechanism that replaces linear projection with a learnable Mahalanobis metric. By enabling each latent slice to learn a local anisotropic tensor, METRO shapes receptive fields into exponentially localized, oriented ellipsoids that naturally align with flow features like boundary layers and wakes. As a drop-in replacement, METRO yields consistent improvements across both Transformer and Mamba backbones. Empirically, our method achieves substantial performance gains on irregular domains, outperforming baselines on both standard PDE benchmarks and complex industrial design tasks. Finally, METRO exhibits enhanced robustness in…