Hyperbolic Neural Operator

Neural operators have emerged as powerful surrogates for solving PDEs, significantly accelerating scientific computation. While transformer-based architectures offer unmatched flexibility for irregular domains, they suffer from a fundamental efficiency gap: standard attention mechanisms assign uniform interaction budgets to all token pairs, neglecting the physical reality that far-field interactions are often compressible. To address this mismatch, we draw inspiration from classical fast solvers that exploit hierarchical near-far decompositions. We further observe that embedding such tree-structured hierarchies in Euclidean space incurs inherent distortion, whereas hyperbolic space naturally accommodates exponential branching. Consequently, we propose \textbf{Hyperbolic Neural Operator (HNO)}, which leverages intrinsic hyperbolic geometry to instantiate a continuous Gibbs kernel based on stabilized geodesic distances on the Lorentz hyperboloid. This design imposes a geometric inductive bias that naturally enjoys robust multi-scale routing akin to the Fast Multipole Method (FMM), yet within a unified, learnable attention mechanism. Empirically, HNO achieves state-of-the-art accuracy on six PDE benchmarks and two large-scale unstructured CFD tasks, reducing the mean relative $\ell_2$ error by up to 40\% compared to leading baselines. Codes are attached and will be available online.

Many problems in science and engineering require computer simulations, such as predicting how fluids move, how materials bend, or how air flows around vehicles. These simulations can be slow, so researchers use AI models to make fast predictions. However, many physical systems contain both small local details and long-range effects, which are hard for a model to capture at the same time. This paper introduces HNO, an AI model designed to better organize these different interaction ranges. The key idea is to use a geometry similar to a branching tree, where nearby and faraway relationships can be represented more naturally. This helps the model decide when to focus on local details and when to use broader information from the whole system. We test HNO on several standard scientific simulation tasks and two large air-flow prediction tasks. It gives more accurate predictions than the compared methods. Additional tests show that the improvement comes from the model’s way of organizing near and far interactions, rather than simply using a larger model.