# Simple yet Effective: Low-Rank Spatial Attention for Neural Operators

arXiv: https://arxiv.org/abs/2604.03582 · HTML: https://mlsanity.com/paper/2604.03582/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 46 references that are in the map.
- First posted on arXiv 4 Apr 2026.

## Builds on (most cited first, 30 of 37)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning](https://mlsanity.com/paper/2307.08691/) (2023)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Perceiver IO: A General Architecture for Structured Inputs & Outputs](https://mlsanity.com/paper/2107.14795/) (2021)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Liger Kernel: Efficient Triton Kernels for LLM Training](https://mlsanity.com/paper/2410.10989/) (2024)
- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [The Devil in Linear Transformer](https://mlsanity.com/paper/2210.10340/) (2022)
- [Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries](https://mlsanity.com/paper/2502.02414/) (2025)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutions](https://mlsanity.com/paper/2212.07564/) (2022)
- [Latent Neural Operator for Solving Forward and Inverse PDE Problems](https://mlsanity.com/paper/2406.03923/) (2024)

## Built on by (most cited first, 4 of 4)

- [From Points to Edges: Edge-Conditioned Spectral Operators for Physics-Sensitive PDE Learning](https://mlsanity.com/paper/2608.06894/) (2026)
- [Learning Laplacian Eigenspace with Mass-Aware Neural Operators on Point Clouds](https://mlsanity.com/paper/2605.24390/) (2026)
- [Read, Write, Relax: Why Neural PDE Surrogates Need Both Global and Local Processing](https://mlsanity.com/paper/2608.21677/) (2026)
- [Does Transolver really need a Transformer?](https://mlsanity.com/paper/2609.32525/) (2026)

## Abstract (arXiv)

Neural operators have emerged as data-driven surrogates for solving partial differential equations (PDEs), and their success hinges on efficiently modeling the long-range, global coupling among spatial points induced by the underlying physics. In many PDE regimes, the induced global interaction kernels are empirically compressible, exhibiting rapid spectral decay that admits low-rank approximations. We leverage this observation to unify representative global mixing modules in neural operators under a shared low-rank template: compressing high-dimensional pointwise features into a compact latent space, processing global interactions within it, and reconstructing the global context back to spatial points. Guided by this view, we introduce Low-Rank Spatial Attention (LRSA) as a clean and direct instantiation of this template. Crucially, unlike prior approaches that often rely on non-standard aggregation or normalization modules, LRSA is built purely from standard Transformer primitives, i.e., attention, normalization, and feed-forward networks, yielding a concise block that is straightforward to implement and directly compatible with hardware-optimized kernels. In our experiments, such a simple construction is sufficient to achieve high accuracy, yielding an average error reduction of over 17% relative to second-best methods, while remaining stable and efficient in mixed-precision training.