Function-Space Transformer with Adaptive Anchors

2026-09-29

AI Summary

This study addresses the inherent trade-off between preserving local details and maintaining computational efficiency when representing functional data on fixed grids. To this end, we propose the FST framework, which introduces a spatially adaptive continuous latent representation to transcend the limitations of conventional observation grids. By leveraging predictive anchor localization and recursive function space interaction techniques, our approach enables efficient learning of features within continuous function spaces. Experimental evaluations demonstrate that the proposed method significantly outperforms existing baselines on both partial differential equation (PDE) solving and ImageNet tasks, achieving superior accuracy while requiring substantially fewer parameters.

Abstract

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.

Problem

• continuous function representation
• neural operators
• spatially adaptive representation
• discrete observations
• latent attention

Innovation

• Function-Space Transformer
• Adaptive Anchors
• Continuous Latent Representation
• Neural Operators
• Recursive Refinement