Diffeomorphic Neural Operator Learning

2025/08/08 by Seth Taylor, Alexander Bihlo, Taylor, Seth +3 · 1 citation
Computer Science · Physics and Astronomy · #Domain Adaptation and Few-Shot Learning #FOS: Mathematics #Generative Adversarial Networks and Image Synthesis #Model Reduction and Neural Networks #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2508.06690

openalex publication_date 2025/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present an operator learning approach for a class of evolution operators using a composition of a learned lift into the space of diffeomorphisms of the domain and the group action on the field space. In turn, this transforms the semigroup structure of the evolution operator into a corresponding group structure allowing time stepping be performed through composition on the space of diffeomorphisms rather than in the field space directly. This results in a number of structure-preserving properties related to preserving a relabelling symmetry of the dynamics as a hard constraint. We study the resolution properties of our approach, along with its connection to the techniques of diffeomorphic image registration. Numerical experiments on forecasting turbulent fluid dynamics are provided, demonstrating its conservative properties, non-diffusivity, and ability to capture anticipated statistical scaling relations at sub-grid scales. Our method provides an example of geometric operator learning and indicates a clear performance benefit from leveraging a priori known infinite-dimensional geometric structure.

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