# Function-Space Transformer with Adaptive Anchors

arXiv: https://arxiv.org/abs/2609.38348 · HTML: https://mlsanity.com/paper/2609.38348/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 28 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 23 of 23)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Perceiver: General Perception with Iterative Attention](https://mlsanity.com/paper/2103.03206/) (2021)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [Universal Transformers](https://mlsanity.com/paper/1807.03819/) (2018)
- [Perceiver IO: A General Architecture for Structured Inputs & Outputs](https://mlsanity.com/paper/2107.14795/) (2021)
- [Learning Continuous Image Representation with Local Implicit Image Function](https://mlsanity.com/paper/2012.09161/) (2020)
- [DeiT III: Revenge of the ViT](https://mlsanity.com/paper/2204.07118/) (2022)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [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)
- [Vision Transformer with Deformable Attention](https://mlsanity.com/paper/2201.00520/) (2022)
- [Hierarchical Reasoning Model](https://mlsanity.com/paper/2506.21734/) (2025)
- [AB-UPT: Scaling Neural CFD Surrogates for High-Fidelity Automotive Aerodynamics Simulations via Anchored-Branched Universal Physics Transformers](https://mlsanity.com/paper/2502.09692/) (2025)
- [CViT: Continuous Vision Transformer for Operator Learning](https://mlsanity.com/paper/2405.13998/) (2024)
- [Continuum Attention for Neural Operators](https://mlsanity.com/paper/2406.06486/) (2024)
- [AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fields](https://mlsanity.com/paper/2406.02176/) (2024)
- [Grounding Continuous Representations in Geometry: Equivariant Neural Fields](https://mlsanity.com/paper/2406.05753/) (2024)
- [Inducing Point Operator Transformer: A Flexible and Scalable Architecture for Solving PDEs](https://mlsanity.com/paper/2312.10975/) (2023)
- [Space-Time Continuous PDE Forecasting using Equivariant Neural Fields](https://mlsanity.com/paper/2406.06660/) (2024)
- [From Basis to Basis: Gaussian Particle Representation for Interpretable PDE Operators](https://mlsanity.com/paper/2602.21551/) (2026)

## Abstract (arXiv)

Many forms of data, including physical fields, geometric shapes, and visual signals, are naturally described by functions over continuous domains but are observed through discrete samples. Representing these functions on fixed uniform grids imposes a trade-off between resolving localized variation and increasing computation across the domain. Neural operators address this mismatch by learning mappings between functions, while latent-attention architectures provide flexible processing of sampled observations. We introduce the Function-Space Transformer (FST), a framework for learning from functions through a spatially adaptive continuous latent representation. FST stores features at anchors whose locations are predicted from the input observations and recursively refines these anchor features through function-space interactions. This allows the representation to adapt its spatial organization to each input rather than inherit that of the observation grid, while supporting both spatially resolved and finite-dimensional outputs. On PDE solution prediction using PDEBench Burgers and Darcy flow, FST substantially outperforms the Perceiver IO baseline, whose latent representation lacks explicit spatial organization, and is highly competitive with the Fourier Neural Operator. On ImageNet-1K, FST achieves higher classification accuracy than the Vision Transformer baseline, with fewer parameters across these comparisons. Ablations further support the benefits of function-space updates and recursive refinement. Together, these results highlight the potential of adaptive continuous representations for both scientific prediction and visual recognition.