Manifold Function Encoder: Identifying Different Functions Defined on Different Manifolds

TL;DR

The paper introduces the Manifold Function Encoder (MFE), a novel spectral method for encoding functions on manifolds, with proven super-algebraic convergence and extensions to complex cases like joint manifolds and PDE solution mappings.

Contribution

The paper presents MFE, a new spectral encoding technique for manifold functions, with theoretical convergence guarantees and applications to PDE operator learning.

Findings

MFE achieves super-algebraic convergence for smooth basis expansions.

MFE effectively encodes manifold functions with complex structures.

Numerical experiments demonstrate MFE's accuracy in PDE problems.

Abstract

We propose the Manifold Function Encoder (MFE) for identifying different functions defined on different manifolds. Both a manifold in Euclidean space and a function defined on this manifold can be viewed as bounded linear functionals on a suitable space of continuous functions. From this perspective, we treat manifold functions as elements of the dual space. By expanding them in the dual space based on appropriate approximating sequence of bases, we obtain a corresponding method for encoding manifold functions, that is MFE. Especially, we prove that MFE achieves super-algebraic convergence based on smooth bases commonly used in spectral methods, such as Legendre polynomials and Fourier basis. We further extend MFE to handle more complex cases, including joint manifold functions of different dimensions and manifold functions with different measures. In addition, we show the approximation theory for MFE-based operator learning, in particular learning the solution mappings of PDEs defined on varying domains, together with several numerical experiments including the 2-d Poisson equation and the 3-d elasticity problem on the real-world bearing.