aiqbench analysis
This research introduces Cluster Attention Neural Operators (CANO), a novel architecture designed to solve parametric partial differential equations with superior speed and accuracy compared to existing Transformer-based methods. By dynamically clustering queries while preserving full-resolution keys and values, CANO overcomes the spatial information loss and flexibility constraints found in prior reduced-space approaches like Transolver. The model demonstrates state-of-the-art performance across complex benchmarks involving fluid dynamics, solid deformation, and irregular geometries, marking a significant step forward in efficient scientific machine learning.
Cluster Attention Neural Operators for Solving Parametric Partial Differential Equations
arXiv:2609.39914v1 Announce Type: cross Abstract: Traditional simulations of parametric partial differential equations (PDEs) rely on repetitive computations for each parameter, which makes high-fidelity design impractical. Neural operators address this issue by learning solution operators, accelerating parameter-space mapping by orders of magnitude. Recent Transformer-based neural operators attempt to capture global dependencies, but often at the cost of quadratic attention complexity. Transolver resolves this problem by projecting physical states into a reduced slice space for attention computation. Although fast, this projection sacrifices fine spatial information. Moreover, by operating in this reduced space with shared weights across attention heads, it may constrain the model's flexibility, thereby limiting its capacity to capture complex phenomena. To address these issues, we propose the Cluster Attention Neural Operator (CANO), which reformulates attention via a novel cross-attention mechanism that dynamically clusters queries while preserving full-resolution keys and values. This avoids slice compression loss and removes weight-sharing limits. At the same time, the model remains fast without losing global interactions. Empirically, CANO achieves state-of-the-art performance across canonical PDE benchmarks, covering fluid and solid dynamics (e.g., Navier-Stokes, Airfoil, Plasticity), irregular unstructured geometries (e.g., Pipe Turbulence, Composites), and long-term temporal rollouts. Across solid deformation and turbulent flow benchmarks, CANO achieves lower errors than baselines and exhibits strong geometric adaptability and temporal consistency.
Abstract:Traditional simulations of parametric partial differential equations (PDEs) rely on repetitive computations for each parameter, which makes high-fidelity design impractical. Neural operators address this issue by learning solution operators, accelerating parameter-space mapping by orders of magnitude. Recent Transformer-based neural operators attempt to capture global dependencies, but often at the cost of quadratic attention complexity. Transolver resolves this problem by projecting physical states into a reduced slice space for attention computation. Although fast, this projection sacrifices fine spatial information. Moreover, by operating in this reduced space with shared weights across attention heads, it may constrain the model's flexibility, thereby limiting its capacity to capture complex phenomena. To address these issues, we propose the Cluster Attention Neural Operator (CANO), which reformulates attention via a novel cross-attention mechanism that dynamically clusters queries while preserving full-resolution keys and values. This avoids slice compression loss and removes weight-sharing limits. At the same time, the model remains fast without losing global interactions. Empirically, CANO achieves state-of-the-art performance across canonical PDE benchmarks, covering fluid and solid dynamics (e.g., Navier-Stokes, Airfoil, Plasticity), irregular unstructured geometries (e.g., Pipe Turbulence, Composites), and long-term temporal rollouts. Across solid deformation and turbulent flow benchmarks, CANO achieves lower errors than baselines and exhibits strong geometric adaptability and temporal consistency.
Mathematical Physics (math-ph); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Dynamical Systems (math.DS)
arXiv:2609.39914 [math-ph]
(or arXiv:2609.39914v1 [math-ph] for this version)
https://doi.org/10.48550/arXiv.2609.39914 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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