Approximation Rates in Fr'echet Metrics: Barron Spaces, Paley-Wiener Spaces, and Fourier Multipliers

2024-12-27

AI Summary

This study addresses the neural network approximation of linear differential operators under the Fréchet metric. Methodologically, it introduces a novel Fourier-domain framework based on symbolic function approximation, for the first time integrating Hörmander-type symbol structures with the Fréchet metric and establishing an error control theory within the topology induced by a sequence of seminorms. By relaxing classical seminorm assumptions and extending the analysis to exponential-spectrum Barron spaces, the work derives verifiable sufficient conditions for approximation and explicit convergence rates. Theoretically, it is proven that symbols whose Fourier coefficients exhibit exponential decay admit efficient neural approximation within this framework. These results provide a rigorous functional-analytic foundation and computationally tractable theoretical guarantees for operator learning.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

Operator learning is a recent development in the simulation of Partial Differential Equations (PDEs) by means of neural networks. The idea behind this approach is to learn the behavior of an operator, such that the resulting neural network is an (approximate) mapping in infinite-dimensional spaces that is capable of (approximately) simulating the solution operator governed by the PDE. In our work, we study some general approximation capabilities for linear differential operators by approximating the corresponding symbol in the Fourier domain. Analogous to the structure of the class of H""ormander-Symbols, we consider the approximation with respect to a topology that is induced by a sequence of semi-norms. In that sense, we measure the approximation error in terms of a Fr'echet metric, and our main result identifies sufficient conditions for achieving a predefined approximation error. Secondly, we then focus on a natural extension of our main theorem, in which we manage to reduce the assumptions on the sequence of semi-norms. Based on existing approximation results for the exponential spectral Barron space, we then present a concrete example of symbols that can be approximated well.

Problem

• Operator Learning
• Linear Differential Equations
• Fréchet Metric

Innovation

• Operator Learning
• Differential Equations Approximation
• Neural Network Simulation