Diffeomorphic Neural Operator Learning

TL;DR

This paper introduces a diffeomorphic neural operator that leverages geometric structure to improve the learning of evolution operators, demonstrating benefits in turbulence forecasting and preserving symmetries.

Contribution

It proposes a novel operator learning framework using diffeomorphisms to encode evolution operators, preserving symmetries and structure in the learned models.

Findings

Demonstrates conservative properties and non-diffusivity in turbulence forecasting.

Captures statistical scaling relations at sub-grid scales.

Shows performance benefits from geometric structure leveraging.

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.