LLT: Local Linear Transformer for PDE Operator Learning
Introduces Local Linear Transformer (LLT), a novel neural operator architecture combining linear global attention with local spatial mixing to address quadratic scaling and lack of local bias in standard attention for PDEs. Achieves competitive errors and 1.8-2.5x training speedup over Transolver on multiple PDE problems.
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[Submitted on 4 Jul 2026]
Title:LLT: Local Linear Transformer for PDE Operator Learning
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Abstract:Neural operators have become a common approach for learning PDE solution maps and accelerating numerical simulations. Transformer-based neural operators are of particular interest, since attention can learn long-range dependencies in the computational domain. However, standard attention has two major limitations when applied to PDEs: it scales quadratically with the number of computational nodes, and it lacks an explicit bias toward local interactions. To address these issues, we introduce Local Linear Transformer (LLT) for PDE operator learning. The architecture combines linear global attention with local spatial mixing, and incorporates coordinate and geometry information. We evaluate LLT on several PDE problems, including elasticity, plasticity, airfoil flow, pipe flow, and Darcy flow. The reference data for these problems span finite-element, finite-volume, and finite-difference discretizations on structured and unstructured meshes. Compared with other neural-operator and transformer baselines from prior studies, LLT achieves competitive or lower relative $L_2$ error across these problems. On matched structured discretizations, wall-clock time per training iteration is reduced by factors of 1.8 to 2.5 relative to Transolver. We also scale the approach and apply it to a three-dimensional car aerodynamics dataset with 32,186 unstructured mesh points per sample. Together, these results indicate that LLT provides an accurate and computationally efficient operator for PDE problems across discretizations, mesh types, and problem settings.
Subjects:
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Numerical Analysis (math.NA)
Cite as: arXiv:2607.07718 [cs.LG]
(or arXiv:2607.07718v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.07718
arXiv-issued DOI via DataCite
Submission history
From: Oded Ovadia [view email] [v1] Sat, 4 Jul 2026 11:07:09 UTC (9,587 KB)
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